❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦
❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤
▼❛✐ ❚❤Õ ❚➞♥
❈➳❝ ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❈❛r✐st✐
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣
▲✉❐♥ ✈➝♥ ❚❤➵❝ sÜ ❚♦➳♥ ❤ä❝
◆❣❤Ö ❆♥ ✲ ✷✵✶✺
❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦
❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤
▼❛✐ ❚❤Õ ❚➞♥
❈➳❝ ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❈❛r✐st✐
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣
▲✉❐♥ ✈➝♥ ❚❤➵❝ sÜ ❚♦➳♥ ❤ä❝
❈❤✉②➟♥ ♥❣➭♥❤✿
❚♦➳♥ ●✐➯✐ tÝ❝❤
▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷
❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝
P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥
◆❣❤Ö ❆♥ ✲ ✷✵✶✺
▼ô❝ ▲ô❝
❚r❛♥❣
▼ô❝ ❧ô❝
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷
▲ê✐ ♥ã✐ ➤➬✉
❈❤➢➡♥❣ ■✳
➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝ r✐➟♥❣
✺
✶✳✶✳
❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥
✶✳✷✳
➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝
r✐➟♥❣
❈❤➢➡♥❣ ■■✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✸
▼ét sè ♠ë ré♥❣ ❝ñ❛ ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❈❛r✐st✐ tr♦♥❣ ❦❤➠♥❣
❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳
✾
➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ❈❛r✐st✐ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝ r✐➟♥❣
✷✳✶✳
✺
✷✸
➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ❈❛r✐st✐ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝ r✐➟♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✻
❑Õt ❧✉❐♥
✸✺
❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦
✸✻
✶
ờ ó
ý tết ể t ộ ột tr ữ ủ ề ứ q
trọ ủ tí ó ó ề ứ ụ tr t ọ ỹ
tt ị ý ể t ộ t ệ từ tế ỉ tr ó
ể ế ý ể t ộ rr ý
ết q ể ợ ở rộ r ớ
t út sự q t ủ ề t
ọ tr ớ
rst ứ ột ị ý ể t ộ ó
ở rộ ủ ý ị ý rst ột tr
ữ ị ý ể t ộ q trọ t tr tr
ủ ở ì ó ò ột ủ ý ế
rt ễ ụ ó rt ề t ọ tì ở rộ ị ý ể
t ộ rst r
ột tr ớ ở rộ ý tì
ở rộ ệ tr tr
tts ớ tệ ệ tr r X
ột ở rộ ủ tr ột tr ữ tí t tú ị ủ
tr r p p(x, x) ó tể 0 ớ x X ó ỗ
tr r ột To r ó tr tts
ứ tí t ủ ộ tụ tr tr r
ũ ứ ột số ị ý ể t ộ s
rộ tr tr r
ột số t ọ tết ột số ớ ủ ị ý
ể t ộ ể rst ú ó tể ợ ụ ể ứ
ột số ở rộ ủ ý tr
tr r
ể t ợt ứ ọ ú t tế ớ
ứ tì ể ết q ề ể t ộ ủ s
rộ ệt ể rst tr tr r
r sở t ệ t ớ sự ớ ủ P r
❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤ù❝ ❤✐Ö♥ ➤Ò t➭✐✿
✧❈➳❝ ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❈❛r✐st✐ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✧✳
▼ô❝ ➤Ý❝❤ ❝ñ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱
♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✱
t➠♣➠ s✐♥❤ ❜ë✐ ♠➟tr✐❝ r✐➟♥❣✱ ❞➲② ✵✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✵✲➤➬②
➤ñ✱ ➤✐Ó♠ ❜✃t ➤é♥❣✱ ➤✐Ò✉ ❦✐Ö♥ ❝♦✱ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ p✲❈❛r✐st✐✱ ➳♥❤ ①➵ ps ✲
❈❛r✐st✐✱ ➳♥❤ ①➵ ❈❛r✐st✐✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ➤✐Ó♠
❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ❈❛r✐st✐✱. . .
❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ò ✧➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✧✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ô❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ö✉ q✉❛
♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ö❝ tr×♥❤ ❜➭② ❝ñ❛ ❧✉❐♥ ✈➝♥✱ ❣å♠✿ ▼➟tr✐❝
r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✱ t➠♣➠ s✐♥❤
❜ë✐ ♠➟tr✐❝ r✐➟♥❣✱ ❞➲② 0✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ➤ñ✱ ➤✐Ó♠
❜✃t ➤é♥❣✱ ➤✐Ò✉ ❦✐Ö♥ ❝♦✱ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ p✲❈❛r✐st✐✱ ➳♥❤ ①➵ ps ✲❈❛r✐st✐✱ ➳♥❤
①➵ ❈❛r✐st✐✳ ❚r×♥❤ ❜➭② ♠ét sè tÝ♥❤ ❝❤✃t ❝ñ❛ ♠➟tr✐❝ r✐➟♥❣ ✈➭ ♠ét sè ➤Þ♥❤ ❧ý
✈Ò ➤✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛
❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ❈❛r✐st✐✳ ▼ô❝ ✷ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ✈Ò ✈✐Ö❝ ♠ë ré♥❣
◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ✈Ò ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝
r✐➟♥❣✳
❈❤➢➡♥❣ ✷ ✈í✐ t➟♥ ❧➭ ✧➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ❈❛r✐st✐ tr♦♥❣
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✧✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ô❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ö✉
tr×♥❤ ❜➭② ♠ét sè ♠ë ré♥❣ ❝ñ❛ ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❈❛r✐st✐ tr♦♥❣ ❦❤➠♥❣
❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ ✈➭ ❝➳❝ ❤Ö q✉➯ ❝ñ❛ ❝❤ó♥❣✳ ▼ô❝ ✷ ❝❤ó♥❣ t➠✐ tr×♥❤
❜➭② ♠ét sè tÝ♥❤ ❝❤✃t ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ❈❛r✐st✐✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t
➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ❈❛r✐st✐ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ ✈➭
❝➳❝ ❤Ö q✉➯ ❝ñ❛ ❝❤ó♥❣✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ò ❝➳❝ ❦Õt q✉➯ ➤ã ✈➭ tr×♥❤ ❜➭②
♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳
▲✉❐♥ ✈➝♥ ♥➭② ➤➢î❝ ❤♦➭♥ t❤➭♥❤ t➵✐ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ❞➢í✐ sù ❤➢í♥❣
❞➱♥ t❐♥ t×♥❤✱ ❝❤✉ ➤➳♦ ✈➭ ♥❣❤✐➟♠ ❦❤➽❝ ❝ñ❛ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ t➳❝ ❣✐➯
①✐♥ ➤➢î❝ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ s➞✉ s➽❝ ➤Õ♥ ❚❤➬②✳
◆❤➞♥ ❞Þ♣ ♥➭② t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ ❇❛♥ ❝❤ñ ♥❤✐Ö♠ ❦❤♦❛
✸
❚♦➳♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ q✉ý ❚❤➬②✱ ❈➠ tr♦♥❣ tæ ●✐➯✐ ❚Ý❝❤ ❦❤♦❛
❚♦➳♥ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ P❤ß♥❣ q✉➯♥ ❧ý ❑❤♦❛ ❤ä❝ ✲ ❙❛✉ ➤➵✐ ❤ä❝ ❚r➢ê♥❣
➜➵✐ ❤ä❝ ❙➭✐ ●ß♥ ➤➲ t➵♦ ➤✐Ò✉ ❦✐Ö♥ ❣✐ó♣ ➤ì tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥
t❤➭♥❤ ❧✉❐♥ ✈➝♥✳
▼➷❝ ❞ï ➤➲ tÝ❝❤ ❝ù❝ ➤➬✉ t➢ ✈➭ ❝ã ♥❤✐Ò✉ ❝è ❣➽♥❣ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉✱ t❤ù❝
❤✐Ö♥ ➤Ò t➭✐✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ ❤➵♥ ❝❤Õ ✈➭ t❤✐Õ✉ sãt✳
❚➳❝ ❣✐➯ ♠♦♥❣ ♥❤❐♥ ➤➢î❝ ♥❤÷♥❣ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ ❝ñ❛ q✉ý ❚❤➬②✱ ❈➠ ✈➭ ❜➵♥
➤ä❝ ➤Ó ❧✉❐♥ ✈➝♥ ➤➢î❝ ❤♦➭♥ t❤✐Ö♥ ❤➡♥✳
❚❤➭♥❤ ♣❤è ❍å ❈❤Ý ▼✐♥❤✱ ♥❣➭② ✸✵ t❤➳♥❣ ✽ ♥➝♠ ✷✵✶✺
▼❛✐ ❚❤Õ ❚➞♥
✹
ể t ộ ủ s rộ
tr tr r
ệ
P ú t ớ tệ q ột số ế tứ sở ệ
trì ủ ộ ồ tr
tr ủ tr r tr r tr
r ủ t p s ở tr r 0 tr
r 0 ủ ể t ộ ề ệ í ụ
ề ó rì ột số tí t ủ tr r ột
số ị ý ề ể t ộ ủ s rộ ột số í ụ
ọ ết q ó
ị ĩ
t ợ X d : X ì X R ợ ọ ột
tr tr X ế tỏ ề ệ
d(x, y) 0 ớ ọ x, y X d(x, y) = 0 ế ỉ ế x = y
d(x, y) = d(y, x) ớ ọ x, y X
d(x, y) d(x, z) + d(z, y) ớ ọ x, y, z X
X ù ớ ột tr d tr ó ợ ọ ột tr
í ệ (X, d) X ố d (x, y) ọ từ ể x
ế ể y
í ụ
ét X = R d : R ì R R ở d (x, y) = |x y| ớ ọ
x, y R ó d ột tr tr R
ét X = Rn ớ t ỳ x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) Rn t t
n
|xi yi |
d1 (x, y) =
tr Rn
1
2
n
2
d2 (x, y) =
i=1
|xi yi | ó d1 , d2 tr
i=1
➜Þ♥❤ ♥❣❤Ü❛✳
✶✳✶✳✸
✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d)✳ ❉➲② {xn } ⊂ X ➤➢î❝
❣ä✐ ❧➭ ❤é✐ tô ✈Ò ➤✐Ó♠ x ∈ X ♥Õ✉ ✈í✐ ♠ä✐ ε > 0 tå♥ t➵✐ n0 ∈ N∗ s❛♦ ❝❤♦ ✈í✐
♠ä✐ n ≥ n0 t❛ ❝ã d (xn , x) < ε✳ ▲ó❝ ➤ã t❛ ❦Ý ❤✐Ö✉ ❧➭ lim xn = x ❤❛② xn → x ❦❤✐
n→∞
n → ∞✳
➜Þ♥❤ ♥❣❤Ü❛✳
✶✳✶✳✹
✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d)✱ {xn } ⊂ X ✳
✶✮ ❉➲② {xn } ⊂ X ➤➢î❝ ❣ä✐ ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈í✐ ♠ä✐ ε > 0✱ tå♥ t➵✐
n0 ∈ N∗ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n, m ≥ n0 t❛ ❝ã d(xn , xm ) < ε✱ ❤❛② {xn } ❧➭ ❞➲② ❈❛✉❝❤②
♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉
lim
n,m→+∞
d(xn , xm ) = 0✳
✷✮ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ➤➢î❝ ❣ä✐ ❧➭ ➤➬② ➤ñ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤②
tr♦♥❣ ♥ã ➤Ò✉ ❤é✐ tô✳
➜Þ♥❤ ♥❣❤Ü❛✳
✶✳✶✳✺
✭❬✶❪✮ ❈❤♦ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ✈➭ (Y, ρ)✳
➳♥❤ ①➵
f : (X, d) → (Y, ρ) ➤➢î❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐ α ∈ [0, 1) s❛♦ ❝❤♦
ρ[f (x) , f (y)] ≤ αd (x, y) ,
➜Þ♥❤ ❧ý✳
✶✳✶✳✻
✈í✐ ♠ä✐ x, y ∈ X.
✭❬✶❪✮ ✭◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦✮ ●✐➯ sö (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝
➤➬② ➤ñ✱ f : X → X ❧➭ ➳♥❤ ①➵ ❝♦ tõ X ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❑❤✐ ➤ã tå♥ t➵✐ ❞✉② ♥❤✃t
➤✐Ó♠ x∗ ∈ X s❛♦ ❝❤♦ f (x∗ ) = x∗ ✳
➜✐Ó♠ x∗ ∈ X ❝ã tÝ♥❤ ❝❤✃t f (x∗ ) = x∗ ➤➢î❝ ❣ä✐ ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ➳♥❤
①➵ f ✳
✶✳✶✳✼
➜Þ♥❤ ♥❣❤Ü❛✳
✭❬✼❪✮ ●✐➯ sö X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✳ ❍➭♠ ψ : X → R
➤➢î❝ ❣ä✐ ❧➭ ♥ö❛ ❧✐➟♥ tô❝ tr➟♥ t➵✐ x0 ∈ X ♥Õ✉ lim sup ψ(x) ≤ ψ(x0 )✳
x→x0
❍➭♠ ψ ➤➢î❝ ❣ä✐ ❧➭ ♥ö❛ ❧✐➟♥ tô❝ tr➟♥ tr➟♥ X ♥Õ✉ ♥ã ❧➭ ♥ö❛ ❧✐➟♥ tô❝ tr➟♥
t➵✐ ♠ä✐ x ∈ X ✳
❍➭♠ ψ ➤➢î❝ ❣ä✐ ❧➭ ♥ö❛ ❧✐➟♥ tô❝ ❞➢í✐ tr➟♥ X ♥Õ✉ ❤➭♠ −ψ ❧➭ ♥ö❛ ❧✐➟♥
tô❝ tr➟♥✱ tr♦♥❣ ➤ã (−ψ)(x) = −ψ(x) ✈í✐ ♠ä✐ x ∈ X ✳
◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ❤➭♠ ψ ➤➢î❝ ❣ä✐ ❧➭ ♥ö❛ ❧✐➟♥ tô❝ ❞➢í✐ t➵✐ x0 ∈ X ♥Õ✉
lim inf ψ(x) ≥ ψ(x0 )✳
x→x0
✻
t ết lim (x) lim (x) ợt t lim sup (x)
xx0
lim inf (x)
xx0
xx0
xx0
ị ĩ
(X, d) tr T : X X
ế tồ t ột ử tụ ớ : X [0, ) tỏ ề ệ
d (x, T x) (x) (T x) , ớ ọ x X,
tì T ọ rst tr (X, d)
ị ý
(X, d) tr ủ số :
X [0, ) ử tụ ớ tỏ ề ệ
d (x, T X) (x) (T x) , ớ ọ x X,
tì T ó ể t ộ tr X
ị ý
tr (X, d) ủ ỉ ỗ
rst ủ tr (X, d) ề ó ột ể t ộ
ị ĩ
X ột t ợ rỗ p : X ì
X R+ ợ ọ ột tr r ptr tr X s ớ ọ
x, y, z X t ó
x = y ỉ p (x, x) = p (x, y) = p (y, y)
p (x, x) p (x, y)
p (x, y) = p (y, x)
p (x, z) p (x, y) + p (y, z) p (y, y)
ó (X, p) ợ ọ ột tr r
ptr
í ụ
X = R+ p : X ìX R+ ị ở p (x, y) = max {x, y}
ớ ọ x, y X ó (X, p) ột tr r
t t t p tỏ ề ệ ủ ị ĩ
t ì trò ủ x, y, z t tí tổ
qt t sử x y z ó t ó max {x, z} max {x, y} + max {y, z}
max {y, y} p (x, z) p (x, y) + p (y, z) p (y, y) ớ ọ x, y, z X p tỏ
ề ệ ủ ị ĩ ó (X, p) ột
tr r
X = R = {x R : x 0} ớ x, y R t ị ĩ p (x, y) =
min {x, y} ó p ột tr r tr R
X = [0, 1] ớ x, y X t ị ĩ p (x, y) = emax{x,y} 1 tì p
ột tr r tr X
ét
ễ t r ế p tr r tr X tì
số ps : X ì X [0, ) ợ ở ps (x, y) = 2p (x, y) p (x, x) p (y, y) ớ ọ
x, y X ột tr tr X
ị ĩ
(X, p) tr r x X
> 0 í ệ Bp (x, ) = {y X : p (x, y) < p (x, x) + } ọ Bp (x, )
ì ở t x í tr tr r (X, p)
ị ý
ợ tt ì ở tr tr
r (X, p) sở ủ ột t p tr X
ứ ễ t r X =
Bp (x, ) sử Bp (x, ) , Bp (y, )
xX,>0
ì ở tù ý tr tr r (X, p) Bp (x, )
Bp (y, ) = ó ớ ỗ z Bp (x, ) Bp (y, ) t ó Bp (z, ) Bp (x, )
Bp (y, ) ớ := p (z, z) + min { p (x, z) , p (y, z)}
ị ý ợ ứ
ị ĩ
(X, p) tr r {xn }
X ó
{xn } ợ ọ ộ tụ tớ ể x X ế p (x, x) = lim p (x, xn )
n
✭✷✮ ❉➲② {xn } ➤➢î❝ ❣ä✐ ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉
lim p (xn , xm ) tå♥ t➵✐ ✈➭ ❤÷✉
n,m→∞
❤➵♥✳
✭✸✮ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ (X, p) ➤➢î❝ ❣ä✐ ❧➭ ➤➬② ➤ñ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤②
{xn } tr♦♥❣ X ➤Ò✉ ❤é✐ tô tí✐ ➤✐Ó♠ x ∈ X s❛♦ ❝❤♦ p (x, x) =
✶✳✶✳✶✼
❇æ ➤Ò✳
lim p (xn , xm )✳
n,m→∞
✭❬✶✶✱ ✶✷❪✮ ❈❤♦ (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✈➭ ❞➲② {xn } ⊂
X ✳ ❑❤✐ ➤ã
✭✶✮ ❉➲② {xn } tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ (X, p) ❧➭ ❞➲② ❈❛✉❝❤② ❦❤✐ ✈➭
❝❤Ø ❦❤✐ ♥ã ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, ps )✳
✭✷✮ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ (X, p) ❧➭ ➤➬② ➤ñ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝ (X, ps ) ❧➭ ➤➬② ➤ñ✳
✶✳✶✳✶✽
➜Þ♥❤ ❧ý✳
✭❬✶✷❪✮ ❈❤♦ (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ A ⊂ X, x0 ∈ X ✳
➜➷t p (x0 , A) = inf {p (x0 , x) : x ∈ A}✳ ❚❛ ❝ã a ∈ A ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ p (a, A) = p (a, a)✳
✶✳✶✳✶✾
➜Þ♥❤ ♥❣❤Ü❛✳
X → X✳
➳♥❤ ①➵ F
✭❬✻❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ➳♥❤ ①➵ F :
➤➢î❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ ♥Õ✉ tå♥ t➵✐ λ ∈ (0, 1) s❛♦
❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X, d (F x, F y) ≤ λm (x, y)✱ tr♦♥❣ ➤ã
m (x, y) = max d (x, y) , d (x, F x) , d (y, F y) ,
1
[d (x, F y) + d (y, F x)] .
2
➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣
✶✳✷
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣
P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ò ✈✐Ö❝ ♠ë ré♥❣ ◆❣✉②➟♥
❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ✈Ò ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳
✶✳✷✳✶
➜Þ♥❤ ❧ý✳
✭❬✹❪✮ ❈❤♦ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ ✈➭
F : X → X ❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦
p (F x, F y) ≤ φ max p (x, y) , p (x, F x) , p (y, F y) ,
✾
1
[p (x, F y) + p (y, F x)]
2
✭✶✳✶✮
ớ ọ x, y X tr ó : [0, ) [0, ) ột tụ
s (t) < t ớ ỗ t > 0 ó F ó ột ể t ộ t
ứ ừ ề ệ ủ ễ t r lim n (t) =
n
0 ớ t > 0 ể t ỳ x0 X ị ột {xn } tr X
ở tứ xn = F xn1 ớ n = 1, 2, ... ế tồ t số n0 ó ớ
n0 = 0, 1, 2, ... tì rõ r xn0 ột ể t ộ ủ F ờ sử
xn = xn+1 ớ ọ n ó từ ề ệ ì p (xn , xn ) + p (xn1 , xn+1 )
p (xn1 , xn ) + p (xn , xn+1 ) t ó
p (xn+1 , xn ) = p (F xn , F xn1 )
(max {p (xn , xn1 ) , p (xn , F xn ) , p (xn1 , F xn1 ) ,
1
2
[p (xn , F xn1 ) + p (xn1 , F xn )]
max p (xn , xn1 ) , p (xn , xn+1 ) , 21 [p (xn1 , xn ) + p (xn , xn+1 )]
(max {p (xn , xn1 ) , p (xn , xn+1 )}) .
ờ ế max {p (xn , xn1 ) , p (xn , xn+1 )} = p (xn , xn+1 ) ớ ột số tự
n ó tì từ t ó p (xn+1 , xn ) (p (xn , xn+1 )) < p (xn+1 , xn )
ề t ì p (xn , xn+1 ) > 0 ó max {p (xn , xn1 ) , p (xn , xn+1 )} =
p (xn , xn1 ) ớ ọ n ó từ t ó p (xn+1 , xn ) (p (xn , xn1 )) . ì tế
p (xn+1 , xn ) n (p (x1 , x0 )) .
t ì max {p (xn , xn ) , p (xn+1 , xn+1 )} p (xn , xn+1 ) , từ t ó
{p (xn , xn ) , p (xn+1 , xn+1 )} n p (x1 , x0 ) .
ó t t ợ
ps (xn , xn1 ) = 2p (xn , xn+1 ) p (xn , xn ) p (xn+1 , xn+1 )
2p (xn , xn+1 ) + p (xn , xn ) + p (xn+1 , xn+1 )
4n (p (x1 , x0 )) .
ề ứ tỏ r lim ps (xn , xn+1 ) = 0 ờ t ó
n
ps (xn+k , xn ) ps (xn+k , xn+k1 ) + ã ã ã + ps (xn+1 , xn )
4n+k1 (p (x1 , x0 )) + ã ã ã + 4n (p (x1 , x0 )) .
➜✐Ò✉ ♥➭② ❝❤ø♥❣ tá r➺♥❣ {xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝ (X, ps )✳ ❱× (X, p) ❧➭ ➤➬② ➤ñ ♥➟♥ tõ ❇æ ➤Ò ✶✳✶✳✶✼ t❛ s✉② r❛ (X, ps ) ❝ò♥❣
➤➬② ➤ñ✱ ✈➭ ✈× t❤Õ ❞➲② {xn } ❤é✐ tô ✈Ò x ∈ X tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, ps )
❤❛② lim ps (xn , x) = 0✳ ❈ò♥❣ tõ ❇æ ➤Ò ✶✳✶✳✶✼ t❛ ❝ã
n→∞
p (x, x) = lim p (xn , x) =
n→∞
lim p (xn , xm ) .
✭✶✳✺✮
n,m→∞
❍➡♥ ♥÷❛✱ ✈× {xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, ps )✱
♥➟♥ t❛ ❝ã
lim p (xn , xm ) = 0 ✈➭ tõ ✭✶✳✹✮ t❛ ❝ã lim p (xn , xn ) = 0✳ ❉♦ ➤ã tõ
n,m→∞
➤Þ♥❤ ♥❣❤Ü❛ ps t❛ ❝ã
n→∞
lim p (xn , xm ) = 0✳ ❱× ✈❐② tõ ✭✶✳✺✮ t❛ ➤➢î❝
n,m→∞
p (x, x) = lim p (xn , x) =
n→∞
lim p (xn , xm ) = 0.
n,m→∞
❇➞② ❣✐ê✱ t❛ ❝❤ø♥❣ tá r➺♥❣ p (x, F x) = 0✳ ●✐➯ sö ♥❣➢î❝ ❧➵✐ ➤✐Ò✉ ♥➭② ❦❤➠♥❣
➤ó♥❣✱ ♥❣❤Ü❛ ❧➭ p (x, F x) > 0✳ ❑❤✐ ➤ã tõ ➤✐Ò✉ ❦✐Ö♥ ❝♦ ✭✶✳✶✮ t❛ t❤✉ ➤➢î❝
p (x, F x) ≤ p (x, F xn ) + p (F xn , F x) − p (F xn , F xn )
≤ p (x, xn+1 ) + p (F xn , F x)
≤ p (x, xn+1 ) + φ (max {p (x, xn ) , p (x, F x) , p (xn , xn+1 ) ,
1
2
[p (x, xn+1 ) + p (xn , F x)]
≤ p (x, xn+1 ) + φ (max {p (x, xn ) , p (x, F x) , p (xn , xn+1 ) ,
1
2
[p (x, xn+1 ) + p (xn , x) + p (x, F x) − p (x, x)]})
= p (x, xn+1 ) + φ (max {p (x, xn ) , p (x, F x) , p (xn , xn+1 ) ,
1
2
[p (x, xn+1 ) + p (xn , x) + p (x, F x)]
.
◆❤ê tÝ♥❤ ❧✐➟♥ tô❝ ❝ñ❛ φ ✈➭ ❝❤♦ n → ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ❝ã
p (x, F x) ≤ φ (p (x, F x))✳ ➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ φ✳ ❱❐②
p (x, F x) = 0 ✈➭ x = F x✳
❇➞② ❣✐ê✱ ❣✐➯ sö z ∈ X ❧➭ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❦❤➳❝ ❝ñ❛ F s❛♦ ❝❤♦ x = z ✳
❑❤✐ ➤ã✱ tõ ➤✐Ò✉ ❦✐Ö♥ ❝♦ ✭✶✳✶✮✱ ✈× p (x, x) = 0✱ t❛ ❝ã
p (x, z) = p (F x, F z)
≤ φ max p (x, z) , p (x, F x) , p (z, F z) , 12 [p (x, F z) , p (z, F x)]
= φ max p (x, z) , p (x, x) , p (z, z) , 12 [p (x, z) , p (z, x)]
= φ (max {p (x, z) , p (z, z)})
= φ (p (x, z)).
➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ φ✳ ❉♦ ➤ã x = z.
✶✶
✶✳✷✳✷
❱Ý ❞ô✳
❈❤♦ X = [0, ∞) ✈➭ p (x, y) = max {x, y}✱ ✈í✐ ♠ä✐ x, y ∈ X ✳ ❉Ô t❤✃②
r➺♥❣ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✳ ❳Ðt ➳♥❤ ①➵ F : X → X
x2
2
t
✈í✐ ♠ä✐ x ∈ X ✈➭ ❤➭♠ φ : [0, ∞) → [0, ∞) ❝❤♦ ❜ë✐ ϕ (t) = 1+t
1+x
✈í✐ ♠ä✐ t ∈ [0, ∞)✳ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ x, y ∈ X ✈í✐ x ≥ y t❛ ❝ã
❝❤♦ ❜ë✐ F x =
p (F x, F y) = max
x2
y2
,
1 + x 1 + y
x2
=
1+x
= φ (p (x, y))
≤ φ max p (x, y) , p (x, F x) , p (y, F y) , 12 [p (x, F y) + p (y, F x)]
.
➜✐Ò✉ ♥➭② ❝❤ø♥❣ tá r➺♥❣ t✃t ❝➯ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ❝ñ❛ ➜Þ♥❤ ❧Ý ✶✳✷✳✶ ➤Ò✉ t❤á❛
♠➲♥✱ ✈➭ ♥❤➢ ✈❐② F ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ X ✳
❚❤Õ ♥❤➢♥❣ t❛ ❦❤➠♥❣ t❤Ó ➳♣ ❞ô♥❣ ❦Õt q✉➯ ❝ñ❛ ▼❛tt❤❡✇s ✭①❡♠ ❍Ö q✉➯
✶✳✷✳✺ ❞➢í✐ ➤➞②✮ ❝❤♦ ✈Ý ❞ô ♥➭②✱ ✈× ❦❤➠♥❣ tå♥ t➵✐ α ∈ (0, 1) s❛♦ ❝❤♦ p (F x, F y) ≤
αp (x, y) ✈í✐ ♠ä✐ x, y ∈ X ✳
❚❤❐t ✈❐②✱ ✈í✐ ❜✃t ❦ú α ∈ (0, 1)✱ ♥Õ✉ t❛ ❧✃② 0 < xα <
t❤× t❛ ❝ã F yα =
yα2
α
1−α
✈➭ 0 < yα ≤ xα ✱
x2α
x2α
✱ p(F xα , F yα ) = max{F xα , F yα } =
✱
1 + xα
1 + xα
p(xα , yα ) = xα ✳ ❑❤✐ ➤ã✱ t❛ ❝ã p(F xα , F yα ) > α.p(xα , yα )✳
1 + yα
≤ F xα =
◆❤❐♥ ①Ðt r➺♥❣ tr♦♥❣ ➜Þ♥❤ ❧ý ✶✳✷✳✶ ♥Õ✉ t❛ ➤➷t φ (t) = λt ✈í✐ ♠ä✐ t ∈ [0, ∞)✱
tr♦♥❣ ➤ã λ ∈ [0, 1)✱ t❤× t❛ ❝ã ❤Ö q✉➯ s❛✉✳
✶✳✷✳✸
❍Ö q✉➯✳
✭❬✹❪✮ ◆Õ✉ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ ✈➭
F : X → X ❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦
p (F x, F y) ≤ λ max p (x, y) , p (x, F x) , p (y, F y) ,
1
[p (x, F y) + p(y, F x)]
2
✈í✐ ♠ä✐ x, y ∈ X ✱ tr♦♥❣ ➤ã λ ∈ [0, 1)✳ t❤× F ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳
✶✳✷✳✹
➜Þ♥❤ ❧ý✳
✭❬✹❪✮ ❈❤♦ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ ✈➭
F : X → X ❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦
p (F x, F y) ≤ ap (x, y) + bp (x, F x) + cp (y, F y) + dp (x, F y) + ep(y, F x)
✶✷
✭✶✳✻✮
✈í✐ ♠ä✐ x, y ∈ X ✱ tr♦♥❣ ➤ã a, b, c, d, e ≥ 0 ✈➭ ♥Õ✉ d ≥ e✱ t❤× a + b + c + d + e < 1✱
♥Õ✉ d < e✱ t❤× a + b + c + d + 2e < 1✳ ❑❤✐ ➤ã✱ F ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳
❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ➤✐Ó♠ ❜✃t ❦ú x0 ∈ X ✳ ❚❛ ①➳❝ ➤Þ♥❤ ♠ét ❞➲② {xn } tr♦♥❣
X ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ xn = F xn−1 ✈í✐ n = 1, 2, ...✳ ◆Õ✉ xn0 = xn0 +1 ✈í✐ ♠ét sè tù
♥❤✐➟♥ n0 ♥➭♦ ➤ã✱ t❤× ❞Ô t❤✃② r➺♥❣ xn0 ❧➭ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ F ✳ ❇➞② ❣✐ê
❣✐➯ sö xn = xn+1 ✈í✐ ♠ä✐ n✳ ❑❤✐ ➤ã tõ ➤✐Ò✉ ❦✐Ö♥ ❝♦ ✭✶✳✻✮ t❛ ❝ã
p (xn+1 , xn ) = p(xn , F xn−1 )
≤ ap (xn , xn−1 ) + bp (xn , F xn ) + cp (xn−1 , F xn−1 ) +
+dp (xn , F xn−1 ) + ep(xn−1 , F xn )
= ap (xn , xn−1 ) + bp (xn , xn+1 ) + cp (xn−1 , xn ) +
+dp (xn , xn ) + ep(xn−1 , xn+1 )
≤ (a + c + e) p (xn , xn−1 ) + (b + e) p (xn , xn+1 ) + (d − e) p(xn , xn ).
✭✶✳✼✮
✲ ◆Õ✉ d ≥ e t❤× tõ ✭✶✳✼✮ t❛ ❝ã
p (xn+1 , xn ) ≤ max
a+c+d a+c+e
,
1−b−e 1−b−d
p (xn , xn−1 )
✭✶✳✽✮
✈í✐ ♠ä✐ n✳
✲ ◆Õ✉ d < e✱ t❤× tõ ✭✶✳✼✮ ❜➺♥❣ ❝➳❝❤ ❜á q✉❛ sè ❤➵♥❣ −ed (xn , xn )✱ t❛ ♥❤❐♥
➤➢î❝
a+c+e
a+c+d+e
,
p(xn , xn−1 ).
1−b−e 1−b−d−e
❉♦ ➤ã tõ ✭✶✳✽✮ ✈➭ ✭✶✳✾✮ t❛ ❝ã p (xn+1 , xn ) ≤ λn p (x1 , x0 )✱ tr♦♥❣ ➤ã
max a+c+d , a+c+e
♥Õ✉ d ≥ e,
1−b−e 1−b−d
λ=
max a+c+d+e , a+c+e
♥Õ✉ d < e.
p (xn+1 , xn ) ≤ max
1−b−e
✭✶✳✾✮
1−b−d−e
❉Ô t❤✃② r➺♥❣ λ ∈ [0, 1)✳ ▼➷t ❦❤➳❝✱ ✈×
max {p (xn , xn ) , p (xn+1 , xn+1 )} ≤ p (xn , xn+1 ) ,
♥➟♥ t❛ ❝ã
max {p (xn , xn ) , p (xn+1 , xn+1 )} ≤ λn p (x1 , x0 ) .
❉♦ ➤ã✱ t❛ ♥❤❐♥ ➤➢➡❝
ps (xn , xn+1 ) = 2p (xn , xn+1 ) − p (xn , xn ) − p (xn+1 , xn+1 )
≤ 2p (xn , xn+1 ) + p (xn , xn ) + p (xn+1 , xn+1 )
≤ 4λn p (x1 , x0 ) .
✶✸
✭✶✳✶✵✮
❱× λ ∈ [0, 1)✱ ➤✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ r➺♥❣ lim ps (xn , xn+1 ) = 0✳ ❇➞② ❣✐ê t❛ ❝ã
n→∞
ps (xn+k , xn ) ≤ ps (xn+k , xn+k−1 ) + . . . . + ps (xn+1 , xn )
≤ 4λn+k−1 p (x1 , x0 ) + . . . . + 4λn p (x1 , x0 )
n
= 4λ
(1−λk )
1−λ p (x1 , x0 )
n
λ
p (x1 , x0 ) .
≤ 4 1−λ
➜✐Ò✉ ♥➭② ❝❤ø♥❣ tá r➺♥❣ {xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝
(X, ps )✳ ❱× (X, p) ❧➭ ➤➬② ➤ñ ♥➟♥ tõ ❇æ ➤Ò ✶✳✶✳✶✼ ❞➲② {xn } ❤é✐ tô ✈Ò ♣❤➬♥ tö
x ∈ X tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, ps )✱ ♥❣❤Ü❛ ❧➭ lim ps (xn , x) = 0✳ ❈ò♥❣ tõ
n→∞
❇æ ➤Ò ✶✳✶✳✶✼ ❝❤ó♥❣ t❛ ➤➢î❝
p (x, x) = lim p (xn , x) =
n→∞
lim p (xn , xm ) .
n,m→∞
✭✶✳✶✶✮
❍➡♥ ♥÷❛ ✈× {xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, ps )✱ t❛ ❝ã
lim ps (xn , xm ) = 0 ✈➭ tõ ✭✶✳✶✵✮ t❛ ❝ã lim p (xn , xn ) = 0✳ ❉♦ ➤ã tõ ➤Þ♥❤ ♥❣❤Ü❛
n,m→∞
ps t❛ ❝ã
n→∞
lim p (xn , xm ) = 0✳ ❱× ✈❐② tõ ✭✶✳✶✶✮ t❛ ♥❤❐♥ ➤➢î❝
n,m→∞
p (x, x) = lim p (xn , x) =
n→∞
lim p (xn , xm ) = 0.
n,m→∞
❇➞② ❣✐ê✱ t❛ ❝❤ø♥❣ tá r➺♥❣ p (x, F x) = 0✳ ●✐➯ sö ➤✐Ò✉ ♥➭② ❦❤➠♥❣ ➤ó♥❣✱
❦❤✐ ➤ã tõ ✭✶✳✻✮ t❛ t❤✉ ➤➢î❝
p (x, F x) ≤ p (x, F xn ) + p (F xn , F x) − p(F xn , F xn )
≤ p (x, xn+1 ) + p (F xn , F x)
≤ p (x, xn+1 ) + ap (x, xn ) + bp (x, F x) + cp (xn , xn+1 ) +
+dp (x, xn+1 ) + ep(xn , F x)
≤ p (x, xn+1 ) + ap (x, xn ) + bp (x, F x) + cp (xn , xn+1 ) +
+dp (x, xn+1 ) + ep (xn , x) + ep (x, F x) .
❈❤♦ n → ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ♥❤❐♥ ➤➢î❝ p (x, F x) ≤ (b + e) p (x, F x)✱
❱× 0 ≤ b + e < 1✱ ➤✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ p (x, F x) = 0 ✈➭ x = F x✳
❚Ý♥❤ ❞✉② ♥❤✃t ❝ñ❛ ➤✐Ó♠ ❜✃t ➤é♥❣ ➤➢î❝ s✉② r❛ ❞Ô ❞➭♥❣ tõ ✭✶✳✻✮✳
❚õ ➜Þ♥❤ ❧ý ✶✳✷✳✹ t❛ ❝ã ❝➳❝ ❤Ö q✉➯ s❛✉✳
✶✹
❍Ö q✉➯✳
✶✳✷✳✺
✭❬✹❪✮✭❇❛♥❛❝❤✮ ❈❤♦ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬②
➤ñ ✈➭ F : X → X ❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦
p (F x, F y) ≤ αp(x, y) ✈í✐ ♠ä✐ x, y ∈ X,
tr♦♥❣ ➤ã 0 ≤ α < 1✳ ❑❤✐ ➤ã F ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳
❍Ö q✉➯✳
✶✳✷✳✻
✭❬✹❪✮✭❑❛♥♥❛♥✮ ❈❤♦ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬②
➤ñ ✈➭ F : X → X ❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦
p (F x, F y) ≤ βp (x, F x) + γp(y, F y) ✈í✐ ♠ä✐ x, y ∈ X,
tr♦♥❣ ➤ã β, γ ≥ 0 ✈➭ β + γ < 1✳ ❑❤✐ ➤ã F ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳
❍Ö q✉➯✳
✶✳✷✳✼
✭❬✹❪✮✭❘❡✐❝❤✮ ❈❤♦ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬②
➤ñ ✈➭ F : X → X ❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦
p (F x, F y) ≤ αp (x, y) + βp (x, F x) + γp(y, F y) ✈í✐ ♠ä✐ x, y ∈ X,
tr♦♥❣ ➤ã α, β, γ ≥ 0 ✈➭ α + β + γ < 1✳ ❑❤✐ ➤ã F ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉②
♥❤✃t✳
✶✳✷✳✽
➜Þ♥❤ ❧ý✳
✭❬✽❪✮ ●✐➯ sö (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✱ α ∈ [0, 1)
✈➭ T : X → X ❧➭ ➳♥❤ ①➵ tõ X ✈➭♦ ❝❤Ý♥❤ ♥ã t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥
p (T x, T y) ≤ max {αp (x, y) , p (x, x) , p (y, y)} ✈í✐ ♠ä✐ x, y ∈ X.
✭✶✳✶✷✮
❑❤✐ ➤ã t❛ ❝ã
✶✮ ❚❐♣ Xp = {x ∈ X : p (x, x) = ρp } ❧➭ ❦❤➳❝ rç♥❣✱ ✈í✐ ρp = inf {p (x, y) : x, y ∈ X}❀
✷✮ ❚å♥ t➵✐ ❞✉② ♥❤✃t ➤✐Ó♠ u ∈ Xp s❛♦ ❝❤♦ T u = u❀
✸✮ ❱í✐ ♠ç✐ x ∈ X ✱ ❞➲② {T n x}n≥1 ❤é✐ tô t❤❡♦ ♠➟tr✐❝ ps tí✐ ➤✐Ó♠ u✳
❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ➤✐Ó♠ ❜✃t ❦ú x ∈ X ✱ ❦❤✐ ➤ã tõ ➤✐Ò✉ ❦✐Ö♥ ❝♦ ✭✶✳✶✷✮ t❛
❝ã p (T x, T x) ≤ max {αp (x, x) , p (x, x)} = p (x, x)✱ ❞➲② {p (T n x, T n x)}n≥0 ❧➭ ❞➲②
❦❤➠♥❣ t➝♥❣ ✈➭
p (T n x, T m x) ≤ max αp T n−1 x, T m−1 x , p T n−1 x, T n−1 x
✶✺
✭✶✳✶✸✮
✈í✐ ♠ä✐ m > n ≥ 1✳ ➜➷t
rx := lim p (T n x, T n x) = inf p (T n x, T n x) ≥ 0
n→∞
n≥1
✈➭
Mx :=
1
p (x, T x) + p (x, x) .
1−α
❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣
p (x, T n x) ≤ Mx ✈í✐ ♠ä✐ n ≥ 0.
✭✶✳✶✹✮
❉Ô ❞➭♥❣ t❤✃② r➺♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✶✹✮ ❧✉➠♥ ➤ó♥❣ ✈í✐ n = 0, 1✳ ●✐➯ sö
r➺♥❣ ✭✶✳✶✹✮ ➤ó♥❣ ✈í✐ n ≤ n0 − 1✱ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ✭✶✳✶✹✮ ➤ó♥❣ ✈í✐ n = n0 ≥ 2✳
❚❤❐t ✈❐②✱ t❛ ❝ã
p (x, T n0 x) ≤ p (x, T x) + p (T x, T n0 x)
≤ p (x, T x) + max αp x, T n0 −1 x , p (x, x) .
✲ ◆Õ✉ max αp(x, T n0 −1 x), p(x, x)
= p(x, x)✱ t❤× t❛ ❝ã
p(x, T x) + max αp(x, T n0 −1 x), p(x, x) = p(x, T x) + p(x, x) ≤
≤
1
p(x, T x) + p(x, x) = Mx .
1−α
✲ ◆Õ✉ max αp(x, T n0 −1 x), p(x, x)
= αp(x, T n0 −1 x)✱ t❤× ♥❤ê ❣✐➯ t❤✐Õt q✉②
♥➵♣ t❛ ❝ã
p(x, T x) + max αp(x, T n0 −1 x), p(x, x) = p(x, T x) + αp(x, T n0 −1 x) ≤
α
p(x, T x) + α.p(x, x) =
1−α
1
1
=
p(x, T x) + α.p(x, x) ≤
p(x, T x) + p(x, x) = Mx .
1−α
1−α
≤ p(x, T x) + αMx = p(x, T x) +
❱× t❤Õ✱ t❛ ❝ã p(x, T n0 x) ≤ Mx ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✶✹✮ ➤➢î❝ ❝❤ø♥❣ ♠✐♥❤✳
❚✐Õ♣ t❤❡♦✱ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣
lim p (T n x, T m x) = rx .
n,m→∞
✭✶✳✶✺✮
❍✐Ó♥ ♥❤✐➟♥✱ tõ ❝➳❝❤ ➤➷t rx ✱ ✈í✐ ♠ä✐ n, m ≥ 1 t❛ ❝ã p (T n x, T m x) ≥ p (T n x, T n x) ≥
rx ✈➭ ✈í✐ ε > 0 ❜✃t ❦× ❝❤♦ tr➢í❝✱ tå♥ t➵✐ sè n0 ≥ 1 s❛♦ ❝❤♦ p (T n0 x, T n0 x) < rx + ε
✶✻
2Mx n0 < rx + ờ ớ ọ n, m 2n0 t ó
p (T n x, T m x) max {p (T n1 x, T m1 x) , p (T n1 x, T n1 x) , p (T m1 x, T m1 x)}
max {2 p (T n2 x, T m2 x) , p (T n2 x, T n2 x) , p (T m2 x, T m2 x)} ...
max {n0 p (T nn0 x, T mn0 x) , p (T nn0 x, T nn0 x) ,
p (T mn0 x, T mn0 x)}
< rx + .
ề é t rx p (T n x, T m x) < rx + ớ ọ n, m 2n0 ó t
ó
lim p (T n x, T m x) = rx ì (X, p) tr r ủ
n,m
tồ t x X s
rx = p (x,
x)
= lim p (x,
T n x) = lim p (T n x, T m x)
n
n,m
sẽ ứ r
p (x,
x)
= p (x,
T x)
.
t ớ ỗ n N t ó
p (x,
T x)
p (x,
T n x) p (T n x, T n x) + p (T x,
T n x) .
ừ ề ệ t s r tồ t ột {nk }k1 số
s p (T x,
T nk x) p x, T nk 1 x , k 1 p(T x,
T nk x)
p (x,
x)
, k 1 p (T x,
T nk x) p T nk 1 x, T nk 1 x , k 1 r ỗ ột
trờ ợ từ t tứ ể q ớ k t
t ợ p (x,
T x)
p (x,
x)
ợ ứ
ờ t ứ r Xp = ừ t p ớ ỗ k N tồ
t xk X s p (xk , xk ) < p + k1 ó t ứ tr ớ ỗ
k N tồ t xk X s p (xk , xk ) = p (xk , T xk ) sẽ ứ r
lim p (x n , x m ) = p .
n,m
t ớ > 0 é tù ý trớ t n0 :=
3
(1)
+ 1 ế k n0 tì
t ó
p p (T x k , T x k ) p (x k , x k ) = rxk p (xk , xk ) < p +
1
1
(1 )
p +
< p +
.
k
n0
3
ì tế t ó
Uk := p (x k , x k ) p (T x k , T x k ) <
(1 )
, ớ ọ k n0 .
3
ó ế k n0 tì p (x k , x k ) = rxk p (xk , xk ) < p + n1 ề é t
0
p (x k , x k ) p +
(1 ) ớ ọ k n0 .
3
ờ ế n, m n0 tì t ó
p (x n , x m ) p (T x n , T x n )+p (T x n , T x m )+p (T x m , T x m )p (T x n , T x n )p (T x m , T x m ) .
ó từ t ó
p (x n , x m ) Un +Um +p (T x n , T x m ) < Un +Um +max {p (x n , x m ) , p (x n , x n ) , p (x m , x m )} .
ì tế sử ụ t t ợ
p p (x n , x m ) max
max
2 2
n , x n ) , 32 (1 ) + p (x m , x m )
3 ; 3 (1 ) + p (x
2
3 ; p + (1 ) < p + ,
ớ ọ m, n n0 ề ứ r
lim p (x n , x m ) = p (X, p)
n,m
tr r ủ tồ t y X s
p (y; y) = lim (y, x n ) = lim p (x n , x m ) = p .
n
n,m
r y Xp ì Xp rỗ
ờ t ỳ x Xp ó từ t ó
p p (T x,
T x)
p (x,
T x)
= p (x,
x)
= r x = p .
r T x = x Xp t từ t s r r {T n x} n1 ộ tụ
t tr ps tớ x
ố ù t ứ r T ó ể t ộ t t
sử u, v Xp ể t ộ ủ T ĩ T u = u, T v = v
ó ờ ề ệ t ó
p (u, v) = p (T u, T v) max {p (u, v) , p (u, u) , p (v, v)} .
ế max {p (u, v) , p (u, u) , p (v, v)} = p (u, v) tì từ t ó p (u, v)
p (u, v) ỉ (1 ) p (u, v) 0 ì [0, 1) ề é t
p (u, v) = 0 r u = v
ế max {p (u, v) , p (u, u) , p (v, v)} = p (u, u) tì từ t s r
p (u, v) p (u, u) ỉ u = v
t t tự trờ ợ max {p (u, v) , p (u, u) , p (v, v)} =
p (v, v) ề ứ tỏ T ó ể t ộ t
ét
ù ị ý t tí t ủ
ể t ộ từ ứ tí t ủ ể t ộ
tr Xp ớ tết ế u v ể t ộ ủ T s
p(u, u) = p(v, v) tì t ó u = v
ế t ề ệ ở ột ề ệ ó tì tí
t ủ ể t ộ ợ
ị ý
sử (X, p) tr r ủ
[0, 1) T : X X tỏ ề ệ
p (T x, T y) max p (x, y) ,
p (x, x) + p (y, y)
2
, ớ ọ x, y X.
ó tồ t ột ể t ộ t z X ữ z Xp ớ
ỗ x Xp {T n x}n1 ộ tụ t tr ps tớ ể t ộ z
ứ ễ t r ế T tỏ ề ệ tì T
tỏ ề ệ ó ó t ị í t s r T ó ể
t ộ ì tế t ỉ ứ r ể t ộ t ế
T z = z T w = w tì ờ ề ệ t ó
p (z, w) = p (T z, T w) max p (z, w) ,
ế max p (z, w) ,
p(z,z)+p(w,w)
2
p (z, z) + p (w, w)
2
.
= p (z, w) tì t ó p (z, w) p (z, w)
ề r ỉ (1 ) p (z, w) 0 ỉ p (z, w) = 0
z = w
p(z,z)+p(w,w)
tì từ t tứ
2
tr t t ợ 2p (z, w) p (z, z) p (w, w) 0 ỉ ps (z, w) 0
ế max p (z, w) ,
p(z,z)+p(w,w)
2
=
ì ps ột tr ps (z, w) = 0 ỉ z = w ề ứ
r T ó ể t ộ t
ừ ị ý tr t t ợ ết q s ủ tts trờ
ợ tr r 0 ủ
ệ q
(X, p) tr r 0 ủ [0, 1)
T : X X từ X í ó tỏ ề ệ
p (T x, T y) p (x, y) ớ ọ x, y X.
ó tồ t t ể t ộ z X ó p (z, z) = 0 ớ ỗ
x X {T n x}n1 ộ tụ t tr ps tớ ể t ộ z
í ụ
X = [0, 1] [2, 3] p : X 2 R ở tứ
max{x, y}
p(x, y) =
|x y|
ế
{x, y} [2, 3] = ,
ế
{x, y} [0, 1].
ó (X, p) ột tr r ủ
t từ t p t ó
x ế x [2, 3]
p(x, x) =
0 ế x [0, 1] ,
y ế y [2, 3]
p(y, y) =
0 ế y [0, 1] ,
max {x, y} ế {x, y} [2, 3] =
p (x, y) =
|x y|
ế {x, y} [0, 1] .
ừ ó t t p (x, x) = p (y, y) = p (x, y) ỉ x = y ĩ
p tỏ ề ệ ủ tr r
ớ {x, y} [0, 1] t ó 0 |x y| p (x, x) p (x, y) ớ {x, y}[2, 3] =
t ó x max {x, y} p (x, x) p (x, y) ó p (x, x) p (x, y) ớ ọ
x, y X p tỏ ề ệ ủ tr r
▼➷t ❦❤➳❝ t❛ ❝ã
max {y, x}
p (y, x) =
|y − x|
max {x, y}
=
|x − y|
♥Õ✉
{y, x} ∩ [2, 3] = φ
♥Õ✉
{y, x} ⊆ [0, 1]
♥Õ✉
{x, y} ∩ [2, 3] = φ
♥Õ✉
{x, y} ⊆ [0, 1]
= p (x, y) .
❙✉② r❛ p (x, y) = p (y, x)✱ ✈í✐ ♠ä✐ x, y ∈ X ✱ ❤❛② p t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ✭✸✮
❝ñ❛ ♠➟tr✐❝ r✐➟♥❣✳
❈✉è✐ ❝ï♥❣✱ ✈í✐ {x, y, z} ⊆ [0, 1]✱ t❛ ❝ã
|x − y| + |y − z| + |y − y| = |x − y| + |y − z| ≥ |x − y + y − z| = |x − z| .
❙✉② r❛ |x − y| + |y − z| + |y − y| ≥ |x − z|✱ ❤❛② p (x, z) ≤ p (x, y) + p (y, z) −
p (y, y)✳ ❱í✐ {x, y, z} ∩ [2, 3] = φ✱ t❛ ❝ã
max {x, y} + max {y, z} − max {y, y} = max {x, y} + max {y, z} − y ≥ max {x, z} ,
❤❛② p (x, z) ≤ p (x, y) + p (y, z) − p (y, y)✳ ❉♦ ➤ã p (x, z) ≤ p (x, y) + p (y, z) − p (y, y)
✈í✐ ♠ä✐ x, y, z ∈ X ✳ ❙✉② r❛ p t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ✭✹✮ ❝ñ❛ ♠➟tr✐❝ r✐➟♥❣✳ ❱❐②
(X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳
❇➞② ❣✐ê ❣✐➯ sö {xn } ❧➭ ❞➲② ❈❛✉❝❤② ❜✃t ❦ú tr♦♥❣ X ✳ ❑❤✐ ➤ã ♥❤ê ❝➳❝❤ ①➳❝
➤Þ♥❤ ❤➭♠ p t❛ s✉② r❛
0
lim p (xn , xm ) =
n,m→∞
x
♥Õ✉
|{xn : n ≥} ∩ [2, 3]|❤÷✉ ❤➵♥
♥Õ✉
|{xn : n ≥} ∩ [2, 3]|✈➠ ❤➵♥.
✈í✐ ♣❤➬♥ tö ♥➭♦ ➤ã x ∈ [2, 3] ⊂ X ✳ ▲➵✐ ♥❤ê ❝➳❝ ①➳❝ ➤Þ♥❤ ❝ñ❛ p✱ t❛ s✉② r❛
lim p (xn , xm ) = p(x, x) ✈í✐ x ∈ X ✳ ❱× t❤Õ ❞➲② {xn } ❤é✐ tô ✈Ò x tr♦♥❣ (X, p) ✈➭
n,m→∞
(X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✳
❚✐Õ♣ t❤❡♦✱ t❛ ①➳❝ ➤Þ♥❤ ➳♥❤ ①➵ T : X → X ❝❤♦ ❜ë✐
x+1
2
Tx =
1
2+x
2
♥Õ✉
0 ≤ x ≤ 1,
♥Õ✉
x = 2,
♥Õ✉
2 < x ≤ 3.
✷✶
❑❤✐ ➤ã t❛ ❝ã
1
p(T x, T y) ≤ p(x, y),
2
♥Õ✉
{x, y} ⊆ [0, 1],
✈➭
p(T x, T y) ≤
p(x, x) + p(y, y)
,
2
♥Õ✉
{x, y} ∩ [2, 3] = φ.
➜✐Ò✉ ♥➭② ❝❤ø♥❣ tá
1
p(x, x) + p(y, y)
p(T x, T y) ≤ max{ p(x, y),
}
2
2
✈í✐ ♠ä✐
x ∈ X.
❉♦ ➤ã✱ ➳♥❤ ①➵ T t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ❝ñ❛ ➜Þ♥❤ ❧ý ✶✳✷✳✶✵ ✈➭ ✈× t❤Õ
T ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❉Ô t❤✃② r➺♥❣ ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ➤ã ❧➭
x = 1✳
❚✉② ♥❤✐➟♥✱ ➳♥❤ ①➵ T ❦❤➠♥❣ t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ✭✶✳✷✹✮ ❝ñ❛ ❍Ö q✉➯
✶✳✷✳✶✶✱ ✈× ♥Õ✉ ❧✃② ❜✃t ❦ú α ∈ [0, 1) t❛ ♥❤❐♥ t❤✃② r➺♥❣
✭❛✮ ◆Õ✉ α ≤ 12 ✱ t❤× ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✷✹✮ ❦❤➠♥❣ t❤á❛ ♠➲♥ ✈í✐ ❜✃t ❦ú x, y ∈
[2, 3)✳
✭❜✮ ◆Õ✉ α ∈ ( 12 , 1)✱ t❤× ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✷✹✮ ❦❤➠♥❣ t❤á❛ ♠➲♥ ✈í✐ ❜✃t ❦ú x, y
2
♥➭♦ ♠➭ 2 < y ≤ x < (2α−1)
✳
❉♦ ➤ã✱ ✈í✐ ❜✃t ❦ú α ∈ [0, 1) ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✷✹✮ ❦❤➠♥❣ t❤á❛ ♠➲♥ ✈í✐
♠ä✐ x ∈ X ✳ ❱× t❤Õ t❛ ❦❤➠♥❣ t❤Ó ➳♣ ❞ô♥❣ ➤➢î❝ ➤Þ♥❤ ❧ý ▼❛tt❤❡✇s ✭❍Ö q✉➯
✶✳✷✳✺✮ ✭t➢➡♥❣ tù ❍Ö q✉➯ ✶✳✷✳✶✶ ❝❤♦ tr➢ê♥❣ ❤î♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬②
➤ñ✮✳
✷✷
❝❤➢➡♥❣ ✷
➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ❈❛r✐st✐
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣
✷✳✶
▼ét sè ♠ë ré♥❣ ❝ñ❛ ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❈❛r✐st✐
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣
P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ♠ë ré♥❣ ❝ñ❛ ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣
❈❛r✐st✐ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ ✈➭ ❝➳❝ ❤Ö q✉➯ ❝ñ❛ ❝❤ó♥❣✳
✷✳✶✳✶
➜Þ♥❤ ❧ý✳
✭❬✷❪✮ ●✐➯ sö (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✱ φ : X →
[0, ∞) ❧➭ ♠ét ❤➭♠ ♥ö❛ ❧✐➟♥ tô❝ ❞➢í✐ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, ps ) ✈➭ t❤á❛
♠➲♥ ➤✐Ò✉ ❦✐Ö♥✿
♥Õ✉
x, y ∈ X
♠➭
p (x, x) = p (x, y)
t❤×
φ (y) ≤ φ (x) ,
✭✷✳✶✮
✈➭ ψ : X → [0, ∞) ❧➭ ❤➭♠ t❤á❛ ♠➲♥ sup{ψ(x) : x ∈ X, φ(x) ≤ inf φ(w) + µ} < ∞
w∈X
✈í✐ µ > 0 ♥➭♦ ➤ã✳ ◆Õ✉ T : X → X ❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥
p (x, T x) ≤ p (x, x) + ψ (x) [φ (x) − φ (T x)] ✈í✐ ♠ä✐ x ∈ X,
✭✷✳✷✮
t❤× T ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ X ✳
❈❤ø♥❣ ♠✐♥❤✳ ❚r♦♥❣ tr➢ê♥❣ ❤î♣ ψ (x) > 0✱ tõ ✭✷✳✷✮ t❛ ❝ã φ (T x) ≤ φ (x) .
❚r♦♥❣ tr➢ê♥❣ ❤î♣ ψ (x) = 0✱ t❛ ❝ã p (x, x) = p (x, T x) ✈➭ ✈× t❤Õ tõ ✭✷✳✶✮ t❛ ❝ã
φ (T x) ≤ φ (x)✳ ❉♦ ➤ã tõ ❧❐♣ ❧✉❐♥ tr➟♥ t❛ ❝ã φ (T x) ≤ φ (x) ✈í✐ ♠ä✐ x ∈ X ✳
➜➷t
Y = {x ∈ X : φ(x) ≤ inf φ(w) + µ} ✈➭ γ = sup ψ (w) < ∞.
w∈X
w∈Y
❱× (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✱ ♥➟♥ tõ ❇æ ➤Ò ✶✳✶✳✶✼ t❛ ❝ã
(X, ps ) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ❝ò♥❣ ✈× φ ❧➭ ❤➭♠ ♥ö❛ ❧✐➟♥ tô❝ ❞➢í✐
tr➟♥ ❦❤➠♥❣ ❣✐❛♥ (X, ps )✱ ♥➟♥ s✉② r❛ Y ❧➭ t❐♣ ❝♦♥ ➤ã♥❣ tr♦♥❣ (X, ps )✳ ❱× t❤Õ
(Y, ps ) ❧➭ ➤➬② ➤ñ✱ ♥➟♥ tõ ❇æ ➤Ò ✶✳✶✳✶✼ t❛ ❝ã (Y, p) ❧➭ ➤➬② ➤ñ✳
✷✸