Về sự tồn tại sóng chạy trong mô hình rời rạc của các quần thể sinh học

➜➵✐ ❤ä❝ q✉è❝ ❣✐❛ ❍➭ ♥é✐

❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❦❤♦❛ ❤ä❝ tù ♥❤✐➟♥

◆❣✉②Ơ♥ ❍÷✉ ❚rÝ

❱Ị sù tå♥ t➵✐
sã♥❣ ❝❤➵② tr♦♥❣ ♠➠ ❤×♥❤ rê✐ r➵❝
❝đ❛ ❝➳❝ q✉➬♥ t❤Ĩ s✐♥❤ ❤ä❝

▲✉❐♥ ✈➝♥ t❤➵❝ sÜ t♦➳♥ ❤ä❝

❍➭ ◆é✐ ✲ ✷✵✶✷


➜➵✐ ❤ä❝ q✉è❝ ❣✐❛ ❍➭ ♥é✐

❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❦❤♦❛ ❤ä❝ tù ♥❤✐➟♥

◆❣✉②Ơ♥ ❍÷✉ ❚rÝ

❱Ị sù tå♥ t➵✐
sã♥❣ ❝❤➵② tr♦♥❣ ♠➠ ❤×♥❤ rê✐ r➵❝
❝đ❛ ❝➳❝ q✉➬♥ t❤Ĩ s✐♥❤ ❤ä❝
❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ❣✐➯✐ tÝ❝❤
▼➲ sè✿ ✻✵ ✹✻ ✵✶✳

▲✉❐♥ ✈➝♥ t❤➵❝ sÜ t♦➳♥ ❤ä❝

◆❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝✿ ❚❙✳ ➜➷♥❣ ❆♥❤ ❚✉✃♥

❍➭ ♥é✐ ✲ ✷✵✶✷


▼ơ❝ ❧ơ❝
▼ơ❝ ❧ơ❝

✐

▲ê✐ ❝➯♠ ➡♥

✐✐

▼ë ➤➬✉

✶

❚ỉ♥❣ ◗✉❛♥

✶

✸

✳ ✳ ✳ ✳

✸

✶✳✷ ❳➞② ❞ù♥❣ ❝➳❝ ➤Þ♥❤ ♥❣❤Ü❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✶ ▼ét sè ♠➠ ❤×♥❤ tr♦♥❣ ❞✐ tr✉②Ị♥ ❤ä❝ ✈➭ t➝♥❣ tr➢ë♥❣ ❞➞♥ sè

✶✳✸ ❍❛✐ ♠Ư♥❤ ➤Ị ❝➡ ❜➯♥
✶✳✹ ❳➞② ❞ù♥❣ tè❝ ➤é sã♥❣

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

✶✶

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

✶✸

❙ù tå♥ t➵✐ ♥❣❤✐Ö♠ sã♥❣ ❝❤➵②

✷

✸✻

✷✳✶ ❚è❝ ➤é ❧❛♥ tr✉②Ò♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✾

✷✳✷ ❙ù ❤é✐ tơ ➤Õ♥ ❣✐➳ trÞ ❝➞♥ ❜➺♥❣

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺

✷✳✸ ❙ù tå♥ t➵✐ ♥❣❤✐Ö♠ sã♥❣ ❝❤➵② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✼

❑Õt ❧✉❐♥

❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦

✻✶

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

✐

✻✷


▼ë ➤➬✉
◗✉➬♥ t❤Ĩ s✐♥❤ ❤ä❝ ❧➭ ♠ét ❤Ư ➤é♥❣ ❧ù❝ tr♦♥❣ t❤ù❝ tÕ ❝ã t➳❝ ➤é♥❣ ❝ñ❛ ❝➳❝
②Õ✉ tè ❦❤➳❝❤ q✉❛♥✳ ❑❤✐ ①❡♠ ①Ðt ♠ét ❤Ư s✐♥❤ t❤➳✐ ❝❤ó♥❣ t❛ ❣➽♥ ♥ã ✈í✐ ♠ét ♠➠
❤×♥❤ t♦➳♥ ❤ä❝ ❝❤♦ ❝➳❝ ❤Ư t❤è♥❣ t✐Õ♥ tr✐Ó♥ t❤❡♦ t❤ê✐ ❣✐❛♥✱ ✈➭ ♥❣➢ê✐ t❛ t❤➢ê♥❣ ❣✐➯
t❤✐Õt ❤Ư t❤è♥❣ ❤♦➵t ➤é♥❣ ❧✐➟♥ tơ❝✱ ❤♦➷❝ rê✐ r➵❝ ➤Ị✉✳ ❚õ ➤ã✱ ❝➳❝ ♣❤Ð♣ tÝ♥❤ ❣✐➯✐ tÝ❝❤
❧✐➟♥ tơ❝ ✈➭ rê✐ r➵❝ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ ➤Ĩ ♠➠ t➯ ❤Ư t❤è♥❣ t➢➡♥❣ ø♥❣ ✈í✐ ❝➳❝ ❣✐➯ t❤✐Õt
t❤ê✐ ❣✐❛♥ ❧ý t➢ë♥❣ ➤➢ỵ❝ ➤➷t r❛✳
❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét ♥❣❤✐➟♥ ❝ø✉ ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠
sã♥❣ ❝❤➵② ❝đ❛ ♠➠ ❤×♥❤ rê✐ r➵❝ tr♦♥❣ ❞✐ tr✉②Ị♥ ❤ä❝ ✈➭ t➝♥❣ tr➢ë♥❣ số
ì ợ rr ứ ❝❤♦ ❦Õt q✉➯ tr♦♥❣ ❜➭✐ ▼❆❚❍✳ ❙■❆▼
❆◆❆▲ ❱♦❧✳ ◆♦✳ ✸✱ ▼❛② ✶✾✽✷✳ ❍✳✧ ▲♦♥❣✲t✐♠❡ ❜❡❤❛✈✐♦r ♦❢ ❛ ❝❧❛ss ♦❢ ❜✐♦❧♦❣✐❝❛❧
♠♦❞❡❧s✧✳
❱í✐ ➤Ị t➭✐✿

❱Ị sù tå♥ t➵✐ sã♥❣ ❝❤➵② tr♦♥❣ ♠➠ ❤×♥❤ rê✐ r➵❝ ❝đ❛ ❝➳❝ q✉➬♥ t❤Ĩ s✐♥❤ ❤ä❝

▲✉❐♥ ✈➝♥ ❣å♠ ✷ ❝❤➢➡♥❣✳
❈❤➢➡♥❣ ✶✳ ❚ỉ♥❣ q✉❛♥✳

◆é✐ ❞✉♥❣ ❝❤➢➡♥❣ ♥➭② ➤➢ỵ❝ ✈✐Õt t❤➭♥❤ ✹ ♠ơ❝✳

▼ơ❝ ✶✳✶ ▼ét sè ♠➠ ❤×♥❤ tr♦♥❣ ❞✐ tr✉②Ị♥ ❤ä❝ ✈➭ t➝♥❣ tr➢ë♥❣ ❞➞♥ sè✳
▼ơ❝ ✶✳✷✳ ❳➞② ❞ù♥❣ ❝➳❝ ➤Þ♥❤ ♥❣❤Ü❛✳
▼ơ❝ ✶✳✸ ❍❛✐ ♠Ư♥❤ ➤Ị ❝➡ ❜➯♥✳
▼ơ❝ ✶✳✹ ❳➞② ❞ù♥❣ tè❝ ➤é sã♥❣✳
❈❤➢➡♥❣ ✷✳ ❙ù tå♥ t➵✐ ♥❣❤✐Ö♠ sã♥❣ ❝❤➵②✳

t❤➭♥❤ ✸ ♠ơ❝✳
▼ơ❝ ✷✳✶ ❚è❝ ➤é ❧❛♥ tr✉②Ị♥
▼ơ❝ ✷✳✷ ❙ù ❤é✐ tơ ➤Õ♥ ❣✐➳ trÞ ❝➞♥ ❜➺♥❣✳
▼ơ❝ ✷✳✷ ❙ù tå♥ t➵✐ ♥❣❤✐Ư♠ sã♥❣ ❝❤➵②✳
❑Õt ❧✉❐♥✳

✶

◆é✐ ❞✉♥❣ ❈❤➢➡♥❣ ✷ ➤➢ỵ❝ ✈✐Õt


❚r♦♥❣ ♣❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ➤➳♥❤ ❣✐➳ ➤ã♥❣ ❣ã♣ ❝đ❛ ❧✉❐♥ ✈➝♥ ✈➭ ➤Ị ❝✃♣ tí✐
❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ t❤ê✐ ❣✐❛♥ t✐Õ♣ t❤❡♦ ➤ã ❧➭ t×♠ ❤✐Ĩ✉ ø♥❣ ❞ơ♥❣ ❧ý t❤✉②Õt ❝đ❛
❲❡♥❜❡r❣❡r ❝❤♦ ❝➳❝ ❧í♣ ♠➠ ❤×♥❤ tr♦♥❣ ➤ã t♦➳♥ tư

Q[u] ❝ã t❤Ĩ ❦❤➠♥❣ ❝♦♠♣❛❝t✳✳

❍➭ ◆é✐✱ ♥❣➭② ✷✺ t❤➳♥❣ ✵✼ ♥➝♠ ✷✵✶✷
❚➳❝ ❣✐➯
◆❣✉②Ơ♥ ❍÷✉ ❚rÝ

✷


❈❤➢➡♥❣ ✶
❚ỉ♥❣ ◗✉❛♥
❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t❛ tr×♥❤ ❜➭② ♠ét số tt ữ ị ĩ
q ế ♠➠ ❤×♥❤ s✐♥❤ t❤➳✐ ✈➭ ❤Ư rê✐ r➵❝ ❝đ❛ ♠ét sè ❝➳❝ q✉➬♥ t❤Ĩ s✐♥❤
❤ä❝✳

✶✳✶

▼ét sè ♠➠ ❤×♥❤ tr♦♥❣ ❞✐ tr✉②Ị♥ ❤ä❝ ✈➭ t➝♥❣
tr➢ë♥❣ ❞➞♥ sè

❈❤ó♥❣ t❛ ①❡♠ ①Ðt ♠ét ì ợ ọ ớ ệ tr trề ❤ä❝
❝đ❛ ♠ét q✉➬♥ t❤Ĩ✳ ❈❤ó♥❣ t❛ ♣❤➞♥ ❧♦➵✐ ❝➳ t❤Ĩ ❝đ❛ q✉➬♥ t❤Ĩ ❝đ❛ ♠ét ❧♦➭✐ ❧➢ì♥❣ ❜é✐
♥❤✃t ➤Þ♥❤✳ ◆Õ✉ ①Ðt ♠ét ❣❡♥ ❣å♠ ❤❛✐ ❛❧❡♥
❦✐Ĩ✉ ❣❡♥✿

AA, Aa, aa✱

❤ỵ♣ tư ❧➭✿

Aa✳

A

✈➭

a✳

❚❤× tr♦♥❣ q✉➬♥ t❤Ĩ sÏ ❝ã ❜❛

tr♦♥❣ ➤ã ❦✐Ĩ✉ ❣❡♥ ồ ợ tử

AA, aa ể ị

trờ sè♥❣ tù ♥❤✐➟♥ ❤♦➷❝ ♥❤➞♥ t➵♦ ➤➢ỵ❝ ♣❤➞♥ ❝❤✐❛ t❤➭♥❤ ❝➳❝ ✈ï♥❣
♣❤➞♥ ❜✐Öt ❣ä✐ ❧➭ ✧ ❍è❝ ✧✳ ❈➳❝ ❝➳ t❤Ĩ ❝đ❛ ❝ï♥❣ ♠ét ❧♦➭✐ sè♥❣ tr♦♥❣ ♠ét ✈ï♥❣ r✐➟♥❣
❜✐Ưt ➤ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét q✉➬♥ t❤Ĩ✳
❈ã sù ❝➳❝❤ ❧② s✐♥❤ s➯♥ ë ♠ét ♠ø❝ ➤é ♥❤✃t ➤Þ♥❤ ✈í✐ ❝➳❝ q✉➬♥ t❤Ó ❧➞♥ ❝❐♥
❝ï♥❣ ❧♦➭✐✳ ❱➭ sù ❞✐ ❝➢✱ ♥❤❐♣ ❝➢ ❝đ❛ ❝➳❝ ❝➳ t❤Ĩ ❧➭♠ t❤❛② ➤ỉ✐ t➬♥ sè ❛❧❡♥✱ ✈➭ t❤➭♥❤

✸


♣❤➬♥ ❦✐Ĩ✉ ❣❡♥ ❝đ❛ q✉➬♥ t❤Ĩ✳
❈➳❝ ❝➳ t❤Ĩ tr♦♥❣ ♠ét q✉➬♥ t❤Ĩ ❣✐❛♦ ♣❤è✐ ♥❣➱✉ ♥❤✐➟♥ ✈í✐ ♥❤❛✉ ➤Ĩ s✐♥❤ r❛
t❤Õ ❤Ư s❛✉✳
❚û ❧Ư sè ❧➢ỵ♥❣ ❛❧❡♥

A ✈í✐ tỉ♥❣ sè ❛❧❡♥ ❝đ❛ ❣❡♥ ➤ã tr♦♥❣ q✉➬♥ t❤Ĩ ➤➢ỵ❝ ❣ä✐

❧➭ t➬♥ sè ❛❧❡♥ ❝ñ❛ ❛❧❡♥ A✱ ❣ä✐ t➬♥ sè ❛❧❡♥ A ë t❤Õ ❤Ư t❤ø n tr♦♥❣ q✉➬♥ t❤Ĩ ❧➭✿
❦❤✐ ➤ã t➬♥ sè ❛❧❡♥ ❝ñ❛ ❛❧❡♥

a ❧➭✿ 1 − un (i)✳

t❤➭♥❤ ♣❤➬♥ ❦✐Ĩ✉ ❣❡♥ t➢➡♥❣ ø♥❣✿

un (i)

❚❤❡♦ ➤Þ♥❤ ❧✉❐t ❍❛r❞②✲ ❲❡✐♥❜❡r❣ t❤×

AA; Aa; aa ❝đ❛ q✉➬♥ t❤Ĩ t➢➡♥❣ ø♥❣ ❧➭

(un (i))2 : 2un (1 − un ) : ((1 − un (i))2
tr♦♥❣ ➤✐Ị✉ ❦✐Ư♥ ❦❤➠♥❣ ❝ã sù t➳❝ ➤é♥❣ ❝đ❛ ❝❤ä♥ ❧ä❝ tù ♥❤✐➟♥✱ ❦❤➠♥❣ ①➯② r❛ ➤ét
❜✐Õ♥ ♠➭ ❝❤Ø ♣❤ô t❤✉é❝ ✈➭♦ ❦✐Ĩ✉ ❣❡♥ ❝đ❛ ♥ã ➤è✐ ✈í✐ ❣❡♥ ➤➢ỵ❝ ①❡♠ ①Ðt✳
❙ù ♣❤➞♥ ➤➠✐ tr♦♥❣ ❣✐❛✐ ➤♦➵♥ ❞✐ ❝➢ ❝ñ❛ ❜❛ ❦✐Ĩ✉ ❣❡♥ ❝ã ❝➳❝ tû ❧Ư

(1 + si ) : 1 : (1 + ti )
s❛✉ ➤ã tû ❧Ö sè♥❣ sãt t➵✐ t❤ê✐ ➤✐Ó♠ ❞✐ ❝➢ ❧➭

(1 + si )(un (i))2 : 2un (1 − un ) : (1 + ti )((1 − un (i))2 .
❈❤ó♥❣ t❛ ❣✐➯ ➤Þ♥❤ r➺♥❣ tỉ♥❣ sè ❝➳❝ ❝➳ t❤Ĩ tr♦♥❣ ❝➳❝ ❝➳ t❤Ĩ ❧✐➟♥ q✉❛♥ ➤Õ♥
sù ♣❤➞♥ ❧♦➭✐ t❤ø ✐ sè♥❣ sãt s❛✉ ❦❤✐ ❞✐ ❝➢ ❧➭ pi ❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ ❦✐Ĩ✉ ❣❡♥ ❝đ❛
❝❤ó♥❣✳ ●✐➯ sư r➺♥❣ lij ❧➭ ♠ét ♣❤➬♥ ❝➳ tể ủ ỗ ể tr tể
q ➤Õ♥ sù ♣❤➞♥ ❧♦➭✐ t❤ø

i ❞✐ ❝➢ trë t❤➭♥❤ ♠ét ♣❤➬♥ ❝đ❛ ❝➳❝ ❝➳ t❤Ĩ ❧✐➟♥ q✉❛♥ ➤Õ♥

sù ♣❤➞♥ ❧♦➭✐ t❤ø j ✳ ❑❤✐ ➤ã ♣❤➬♥ ❣❡♥ tr♦♥❣ ❝➳❝ ❝➳ t❤Ó ❧✐➟♥ q✉❛♥ ➤Õ♥ sù ♣❤➞♥ ❧♦➭✐
t❤ø

j

s❛✉ ❦❤✐ ❞✐ ❝➢ ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝✿

un+1 (j) =

mji gi (un (i))

✭✶✳✶✳✶✮

i

tr♦♥❣ ➤ã✿

gi (u) =

2(1 + si )u2 + 2u(1 − u)
2[(1 + si )u2 + 2u(1 − u) + (1 + σi )(1 − u)2 ]
✹

✭✶✳✶✳✷✮


❧➭ ♣❤➬♥ ❣❡♥ ❝đ❛ ❞❡♠❡ t❤ø ✐ tr➢í❝ t❤ê✐ ➤✐Ĩ♠ ❞✐ ❝➢ ✈➭

lji pi
k lji pi

mji =
❧➭ ♠ét ♣❤➬♥ ❝➳❝ ❝➳ t❤Ĩ ❝đ❛ ❞❡♠❡ t❤ø
❑❤✐ ➤ã ❤➭♠

j

✭✶✳✶✳✸✮

❞✐ ❝➢ ➤Õ♥ ❤è❝ t❤Ý❝❤ ❤ỵ♣ ✈➭♦ ➤ê✐ t❤ø i✳

{un (i) : i = 1; 2; ...} t❤á❛ ♠➲♥ ❤Ư
un+1 = Q[un ], (1)

✈í✐

Q[u](j ) =

✭✶✳✶✳✹✮

mji gi (un (i)).
i

❈➠♥❣ ✈✐Ư❝ ❤✐Ư♥ t➵✐ ❝❤ó♥❣ t❛ ❝❤Ø ①Ðt ✈í✐ ♠➠✐ tr➢ê♥❣ sè♥❣ ➤å♥❣ ♥❤✃t✳ ❇➺♥❣
❝➳❝❤ ♥➭②✱ t❛ ①❡♠ ①Ðt t✃t ❝➯ ❝➳❝ ✧❍è❝✧ ❣✐è♥❣ ♥❤❛✉ ♥❤❐♥ ợ ết q
tị tế ế tờ ể tí ợ ớ ệ số ị ể lij ù ợ ♣❤ơ t❤✉é❝
✈➭♦ t❤Õ ❤Ư t❤ø

i t❤Ý❝❤ ❤ỵ♣ ❝❤♦ t❤Õ ❤Ư t❤ø j ✳

▼ét tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt✱ r➺♥❣ ❝➳❝ s✐♥❤ ✈❐t ➤❛♥❣ sè♥❣ tr♦♥❣ ♠➷t ♣❤➻♥❣

R2 ✳

❇✐Ĩ✉ ❞✐Ơ♥ ❜ë✐ ❜➯♥ ➤å ❝❤✐❛ t❤➭♥❤ ❝➳❝ ➠ ✈✉➠♥❣

1
1
1
1
{(x; y)|(k − )h < x < (k + )h, (l − )h < y < (l + )h, k, l = 0; 1; 2...}
2
2
2
2
✈í✐ ➤é ❞➭✐

h✱

➤ã ❧➭ ❝➳❝ ✧❍è❝✧✳ ❚ä❛ ➤é t➞♠ ❝đ❛ ❤×♥❤ ✈✉➠♥❣ ❧➭ ❜é✐ ❝đ❛

♥❤➢ ♠ét ✈❡❝t♦r✳
❝đ❛

h✳

h

✈➭ ①❡♠

❝ã t❤Ĩ ①➳❝ ➤Þ♥❤ tr➟♥ ❝➡ së tõ ❤❛✐ ✈❡❝t♦r ❝ã t❤➭♥❤ ♣❤➬♥ ❧➭ ❜é✐

H

❚r➟♥ t❤ù❝ tÕ

H

t❤ù❝ ❝❤✃t ❧➭ sù ➤å♥❣ ♥❤✃t

s ✈➭ t✱ p ❝➳❝ ❞➵♥❣ tr➢ë♥❣ t❤➭♥❤

➤Ò✉ ❣✐è♥❣ ♥❤❛✉ ë t✃t ❝➯ ❝➳❝ ✧❍è❝✧✱ ✈➭ ❤Ư sè ❞Þ❝❤ ❝❤✉②Ĩ♥ lij ❝❤Ø ♣❤ơ tộ
sự ệt ủ tr
ị r

xi xj

ữ tr t➞♠ ❝đ❛ ❝➳❝ ✧❍è❝✧✳ ❍✐Ư♥ t➵✐ t❛ ❝❤Ø ❣✐➯

s, t, p ❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ u✱ ❞♦ ➤ã ❝❤ó♥❣ ❧➭ ♠ét ❤➺♥❣ sè✳ ❙❛♦ ❝❤♦
l(xi − xj ) =

lij =
k

l(xi ) =
i

k

❚õ ✭✶✳✸✮ t❛ ❝ã

mij = lij ≡ m(xi − xj ).

✺

li0 = 1.
i


❚❛ ➤Þ♥❤ ♥❣❤Ü❛ t♦➳♥ tư ◗ ♥❤➢ s❛✉✿

✭✶✳✶✳✺✮

m(x − y)g(u(y)),

Q[u](x) =
y∈H

tr♦♥❣ ➤ã✿

mji (x) = 1
y∈H

✈➭

g(u) =

su2 + u
.
1 + su2 + σ(1 − u)2

✭✶✳✶✳✻✮

u(1 − u)[su − σ(1 − u)]
.
1 + su2 + σ(1 − u)2

✭✶✳✶✳✼✮

❈❤ó♥❣ t❛ t❤✃② r➺♥❣✿

g(u) − u =

➜Þ♥❤ ♥❣❤Ü❛ ❝đ❛ u ❝❤♦ t❤✃② r➺♥❣ t❛ ❝❤Ø ♣❤➯✐ ①Ðt ❤➭♠ u(x) s❛♦ ❝❤♦ 0
❉Ô ❞➭♥❣ ♥❤❐♥ t❤✃② r➺♥❣

g

t➝♥❣ tõ

0

➤Õ♥

♥❤❐♥ ❣✐➳ trÞ tr♦♥❣ ➤♦➵♥

[0; 1]✳

❚õ ✭✶✳✼✮ t❛ t❤✃② r➺♥❣

❝ã ❜❛ tÝ♥❤ ❝❤✃t✿

✭✐✮ ◆Õ✉

g

1

❣✐è♥❣ ♥❤➢

u

t➝♥❣ tõ

0

➤Õ♥

1

u

1✳

✈➭

Q[u]

s > 0 > σ ✱ ➤å♥❣ ❤ỵ♣ tư AA ❧➭ ♣❤ï ❤ỵ♣ ✈í✐ ➤å♥❣ ❤ỵ♣ tư aa ✈➭
g(u) > u, ✈í✐ 0 < u < 1.

ề ó ò t tr trờ ợ ị ợ tử tr ù tr
ế

s < 0 < σ✱

♥Õ✉ t❤❛② t❤Õ ❜✐Õ♥
✭✐✐✮ ◆Õ✉

s ✈➭ σ

t❤×

g(u) < u,

0 < u < 1✳

❚❛ ❝ã t❤Ĩ ❧♦➵✐ ❜á tr➢ê♥❣ ❤ỵ♣ tr➟♥

u t❤➭♥❤ 1 − u✱ ✈➭ t❤❛② ➤ỉ✐ t❤✉é❝ tÝ♥❤ ❝đ❛ A ✈➭ a✳✮

❧➭ ❝➳❝ sè ➞♠✳ ❚õ ✭✶✳✶✳✼✮ t❛ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t✿

g(u) > u, ✈í✐ 0 < u < π1 ,

g(u) < u, ✈í✐ π1 < u < 1.

✻

✭✶✳✶✳✾✮

❚r♦♥❣ ➤ã

σ
.
s+σ

π1 =
✭✐✐✐✮ ❑❤✐ s ✈➭

σ

✭✶✳✶✳✶✵✮

❧➭ ❝➳❝ sè ❞➢➡♥❣✱ ❦❤✐ ➤ã✿

g(u) < u, ✈í✐ 0 < u < π0 ,

g(u) > u, ✈í✐ π0 < u < 1.

✭✶✳✶✳✶✶✮

❚r♦♥❣ ➤ã✿

π0 =

σ
.
s+σ

❈❤ó ý r➺♥❣ ❝❤➢❛ ❝ã ❧ý ❞♦ ❝ơ t❤Ĩ ➤Ĩ

1+s

✭✶✳✶✳✶✵✬✮
✈➭

1+σ

❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦

u ❦❤✐ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❞➞♥ sè ❝➵♥❤ tr❛♥❤✳ ◆Õ✉ p ❧➭ ❤➺♥❣ sè ✈➱♥ ❝ã ➤➢ỵ❝ ❝➳❝ ❝➠♥❣
t❤ø❝ ✭✶✳✺✮❀ ✭✶✳✻✮✱ ❝➠♥❣ t❤ø❝ ✭✶✳✶✳✼✮ ❝❤♦ t❤✃② r➺♥❣ t❤❐♠ ❝❤Ý
✈➭♦

s ✈➭ σ

✈➱♥ ♣❤ô t❤✉é❝

u ✈➭ ❦❤✐ ➤ã ✭✶✳✶✳✽✮ ✈➱♥ t❤á❛ ♠➲♥✳
◗✉② ♠➠ ❞➞♥ sè ❝ì

p tr➢í❝ ❦❤✐ ❝❤✉②Ĩ♥ ➤ỉ✐ ❝ã t❤Ĩ ♣❤ơ t❤✉é❝ ✈➭♦ t❤➭♥❤ ♣❤➬♥

❝✃✉ t➵♦ ❞✐ tr✉②Ị♥ ❝đ❛ ❞➞♥ sè ✈➭ ❞♦ ➤ã tr➟♥

u✳ ❚õ ✭✶✳✶✳✸✮ ❝❤♦ t❤✃② r➺♥❣ tr♦♥❣ ♠ét

♠➠✐ tr➢ê♥❣ ➤å♥❣ ♥❤✃t ♠ij ❧➭ ♠ét ❤➭♠ ❝đ❛ ✉✭①i ✮ ❝ị♥❣ ♥❤➢ ❝đ❛

xi − xj ✳ ◆ã✐ ❝❤✉♥❣

❝➳❝ ♠➠ ❤×♥❤ ❞✐ ❝❤ó ❝ã t❤Ĩ ♣❤ơ t❤✉é❝ ✈➭♦ ❦✐Ĩ✉ ❞✐ tr✉②Ị♥ ✈➭ ❦❤➯ ♥➝♥❣ s✐♥❤ s➯♥✳
◆Õ✉ t❛ ❣✐➯ ➤Þ♥❤ ❦❤➠♥❣ ❝ã sù t➳❝ ➤é♥❣ s❛✉ ❞✐ ❝➢✱ ❦❤✐ ó t ó t ợ
trị lA (x
ố t➵✐

x

− y, u)

❝đ❛ ❣✐❛♦ tư

A

✈➭ la (x

❜ë✐ ♠ét ❞❡♠❡ s✐♥❤ r❛ t➵✐

y

− y, u)

❝đ❛ ♠ét ❣✐❛♦ tư s✐♥❤ r❛ tr♦♥❣

♥Õ✉ ♣❤➬♥ ❣❡♥ ❜❛♥ ➤➬✉ ❧➭

u✳

❚r♦♥❣ tr➢ê♥❣

❤ỵ♣ ♥➭② t❛ ❝ã ❤Ư

un+1 = Q[un ], (1)
✈í✐ t♦➳♥ tư

Q[u](x) =

− y, u(y))
.
y∈H [lA (x − y, u(y)) − la (x − y, u(y))]
y∈H lA (x

✼

✭✶✳✶✳✶✷✮


◆Õ✉ ❝ã ❣✐❛♦ ♣❤è✐ ❦❤➠♥❣ ♥❣➱✉ ♥❤✐➟♥✱ ♣❤➬♥ ❣❡♥ ♠í✐ ❧➭ ♠ét ❤➭♠ ♥❤➢ ✈Õ ♣❤➯✐ ❝đ❛
✭✶✳✶✳✶✷✮✳
➜✐Ị✉ ❦✐Ư♥ tr➟♥ ❝ã t❤Ĩ sư❛ ➤ỉ✐ t❤❡♦ ❝➳❝ ❝➳❝❤ ❦❤➳❝ ♥❤❛✉✳ ❈➳❝ ❤è❝ ♥➭② ❝ã t❤Ĩ
❜❛♦ ❣å♠ ❤×♥❤ ❜×♥❤ ❤➭♥❤✱ ❤×♥❤ ❧ơ❝ ❣✐➳❝✱ ❤♦➷❝ ❦❤✉ ✈ù❝ ❦❤➳❝ t❤❛② ✈× ❤×♥❤ ✈✉➠♥❣✳
❈❤ó♥❣ t❛ ❝ã t❤Ĩ ①❡♠ ❣✐í✐ ❤➵♥ ❣✐÷❛ ❝➳❝ ✈ï♥❣ ❧➭ r✃t ♥❤á✱ ❞♦ ➤ã
❝➳❝ ❤➭♠ ❧✐➟♥ tô❝ t❤❡♦ ❜✐Õ♥

un (x) sÏ trë t❤➭♥❤

x ✈➭ tr♦♥❣ ❝➠♥❣ t❤ø❝ ✭✶✳✺✮ t♦➳♥ tö Q t❤❛② ❜ë✐
✭✶✳✶✳✶✸✮

m(x − y)g(u(y)).

Q[u](x) =
R2

◆Õ✉

m(x)

❝❤Ø ♣❤ô t❤✉é❝ ✈➭♦ ❦❤♦➯♥❣ ❝➳❝❤ ❊✉❝❧✐❞

|x|✳

❚r♦♥❣ ♠➠✐ tr➢ê♥❣

sè♥❣ rê✐ r➵❝ ✭❦❤➠♥❣ ❧✉➞♥ ♣❤✐➟♥ ➤è✐ ①ø♥❣✮✱ ❦❤✐ ➤ã t❛ t❤❛② t❤Õ t♦➳♥ tư

Q ➤Ĩ ❝ã ➤➢ỵ❝

u ✈í✐ ❝➳❝ ♠è✐ ❧✐➟♥ ❤Ư ✈Ị ➤✐Ị✉ ❦✐Ư♥ ❜❛♥ ➤➬✉✳
◆Õ✉ t❛ ➤Þ♥❤ ♥❣❤Ü❛

un (x) = u(nτ, x)✱

tr♦♥❣ ➤ã

u(t, x)

❧➭ ♥❣❤✐Ư♠ ❝đ❛

♣❤➢➡♥❣ tr×♥❤

∂u
= D∆u + f (u).
∂t
❑❤✐ ➤ã

un

✭✯✮

t❤á❛ ♠➲♥

un+1 = Q[un ], (1)
tr ó

Q[v](x)

ợ ị ĩ

u(, x)

ớ

u(t, x)

ệ ủ

trì

u
= D∆u + f (u),
∂t

✶✳✷

u(0, x) = v(x).

✭✶✳✶✳✶✹✮

❳➞② ❞ù♥❣ ❝➳❝ ➤Þ♥❤ ♥❣❤Ü❛

❚r♦♥❣ ♣❤➬♥ ♥➭② t❛ ①➞② ❞ù♥❣ ♠ét sè ➤Þ♥❤ ♥❣❤Ü❛ ✈Ị ❝➳❝ ❤Ư s✐♥❤ t❤➳✐ ✈➭ ❝➳❝
❝➠♥❣ t❤ø❝ t♦➳♥ ❤ä❝ ❝đ❛ ♥ã✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳

tr♦♥❣

Rn

▼ét ♠➠✐ tr➢ê♥❣ sè♥❣

✈í✐ ❝➳❝ tÝ♥❤ ❝❤✃t s ế

H

ợ ị ĩ t ể

x, y ∈ H
✽

t❤×

x + y, x − y ∈ H ✳


◆Õ✉
◆❤➢ ✈❐②

H

x, y ∈ H

t❤×

x + y, x − y ∈ H ✱

➤✐Ị✉ ♥➭② ❝❤ø♥❣ tá ♣❤➬♥ tư

0 ∈ H✳

❧➭ ♠ét ♥❤ã♠ ➤è✐ ✈í✐ ♣❤Ð♣ ❝é♥❣✳

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳

●ä✐

B

❧➭ t❐♣ ❝➳❝ ❤➭♠ sè ❧✐➟♥ tô❝ tr➟♥

H

[0; π+ ]✱✭ ♥Õ✉ π+ = ∞ t❤× B ❧➭ t❐♣ ❝➳❝ ❤➭♠ sè ❧✐➟♥ tơ❝ tr➟♥ H

♥❤❐♥ ❣✐➳ trÞ tr➟♥
♥❤❐♥ ❣✐➳ trÞ tr➟♥

[0; ∞] ✮✳
❱í✐ ỗ

y

tộ

H

t ị ĩ t tử ị ể

Ty u(x) u(x y).
ột ệ tế ó ợ ị t q ❧✉❐t

un+1 = Q[un ],
tr♦♥❣ ➤ã

un+1

✈➭

un

❧➭ ❝➳❝ ♣❤➬♥ tư ❝đ❛

B ✈➭ Q ❧➭ ♠ét t♦➳♥ tư tr♦♥❣ B✳ ❈➳❝ tÝ♥❤

❝❤✃t ❝đ❛ ❤Ư t✐Õ♥ ❤ã❛ ❧➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛

Q✳

❚õ ❝➳❝ ♥❤❐♥ ①Ðt tr➟♥ t❛ ❣✐➯ t❤✐Õt t♦➳♥ tö
❚♦➳♥ tö

Q ♥❤➢ s❛✉✿

Q ❧➭ ❧✐➟♥ tơ❝ ✈í✐ sù t❤❛② ➤ỉ✐ ❝đ❛ u✱ ✈➭ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t✿

(i) Q[u] ∈ B✱ ∀✉∈ B✳
(ii) Q[Ty [u]] = Ty [Q[u]]✱ ∀✉∈ B✱ ②∈❍✳
(iii)

❈➳❝ ❤➺♥❣

0

π0 < π1

π+

s❛♦ ❝❤♦

Q[α] > α α ∈ (π0 ; π1 ), Q[π0 ] = π0 , Q[π1 ] = π1 , π1 < ∞.
(iv) u

v

s✉② r❛

Q[u]

Q[v]✳

(v) un → u ❦❤✐ n → ∞ ❧➭ ❤é✐ tơ ➤Ị✉ tr➟♥ t❐♣ ❜Þ ❝❤➷♥ tr♦♥❣ H ó
Q[un ](x) Q[u](x), ớ ỗ x ∈ H.

✾

✭✶✳✷✳✶✮


◆❣♦➭✐ r❛ t♦➳♥ tư

Q ❝ß♥ t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿
α ∈ (π1 , π+ ) ♥Õ✉ π1 < π+ .

Q[α] < α,

❍❛✐ t❐♣ ❧å✐ K1 ✈➭ K2 tr♦♥❣ Rn ✱ ✈➭ ♠ét sè ε1
✈➭ ♥Õ✉
✈➭

π1 = π+

u < π1

tr♦♥❣

> 0✳ ●✐➯ sư r➺♥❣ ♥Õ✉ π1 < π+

✈➭ ❝ï♥❣ ✈í✐ ❝➳❝ tÝ♥❤ ❝❤✃t tr➟♥✱ ❦❤✐ ➤ã ♥Õ✉

K1

❤♦➷❝

K2

✭✶✳✷✳✷✮

u < π1 + ε1

✭✶✳✷✳✸✮

b > 0 s❛♦ ❝❤♦
u(x) = 0, ✈í✐ |x|

▼ä✐ ❞➲② ❤➭♠

vn

tr♦♥❣

B

✈í✐

✭✶✳✷✳✹✮

b ⇒ Q[u](0) = 0.

vn

π1

❝ã ♠ét ❞➲② ❝♦♥

vnk

Q[vnk ] ❤é✐ tơ ➤Ị✉ tr➟♥ ♠ä✐ t❐♣ ❜Þ ❝❤➷♥ tr♦♥❣ H ✳
❈❤ó♥❣ t t ớ
ợ

un

H

tì t ó

Q[u](0) < π1 .
❱í✐ ❤➺♥❣ sè

tr♦♥❣

u0

tr♦♥❣

B

s❛♦ ❝❤♦ ❞➲②
✭✶✳✷✳✺✮

❦❤✐ ➤ã sÏ ❞ù ➤♦➳♥ ớ

n

ủ ớ t

ợ ị t q t

un+1 = Q[un ].
❈❤ó♥❣ t❛ ❝❤Ø q✉❛♥ t➞♠ ➤Õ♥ ♥❣❤✐Ư♠ sã♥❣ ❝❤➵② ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ❞➢í✐ ❞➵♥❣

un (x) = W [x · ξ − nc],
tr♦♥❣ ➤ã

W

❦❤➠♥❣ t❤❡♦ q✉② ❧✉❐t ♠ét ❤➭♠ sè ❝ñ❛ ế

ột tr ố

ị ợ ọ só ợ ọ tố ộ só
ị ĩ trị t

W (s)

ợ

s ớ x à nc ✈í✐ x ∈ H, n = 0, 1, 2, ...

❚❛ t❤✃② r➺♥❣

Q[u](x0 ) = Q[T−x0 [u]](0), ∀x0 ∈ Rn .
❍❛✐ tÝ♥❤ ❝❤✃t ❝đ❛ t♦➳♥ tư

Q s✉② r❛ tõ tõ ❝➳❝ ❣✐➯ t❤✐Õt tr➟♥✳
✶✵

✭✶✳✷✳✻✮


✭✈✐✮ ◆Õ✉

α ❧➭ ♠ét ❤➭♠ ❤➺♥❣ ❦❤✐ ➤ã Q[α] ❝ò♥❣ ❧➭ ♠ét ❤➭♠ ❤➺♥❣✳

❚❤❐t ✈❐②✿

α ❧➭ ❤➭♠ ❤➺♥❣ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐

α(x − y) = α(x) = α

✈í✐ ♠ä✐

y ∈ H✳

❚❛ ❝ã

Ty [Q[α]](x) = Q[α](x − y).

▼➭

Q[Ty [α]](x) = Q[α(x − y)](x) = Q[α](x),
✈í✐ ♠ä✐

y ∈ H✳

❚õ

Ty [Q[α]](x) = Q[Ty [α]](x),
s✉② r❛

Q[α](x − y) = Q[α](x) ✈í✐ ♠ä✐ ②.
❱❐②

Q[α] ❧➭ ❤➭♠ ❤➺♥❣✳
✭✈✐✐✮ ❉➲② sè

➤Õ♥ ❤➭♠ ❤➺♥❣

α✳

αn

❤é✐ tơ ➤Õ♥

α ➤➢ỵ❝ ①❡♠ ♥❤➢ ♠ét ❞➲② ❤➭♠ ❤➺♥❣ ❤é✐ tơ ➤Ị✉

◗✉❛ t➳❝ ➤é♥❣ ❝đ❛

Q

t❤× ❞➲② ❤➭♠ ❤➺♥❣

Q[αn ](x)

❤é✐ tơ ➤Õ♥

Q[α].

✶✳✸

❍❛✐ ♠Ư♥❤ ➤Ị ❝➡ ❜➯♥

❚r♦♥❣ ♣❤➬♥ ♥➭② ❝❤ó♥❣ t❛ ➤➢❛ r❛ ❤❛✐ ➤Ị ①✉✃t ❝➡ ❜➯♥ ➤➢ỵ❝ sư ❞ơ♥❣ tr♦♥❣ ❝➳❝
❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ♣❤➬♥ s❛✉✳

✶✶

▼Ư♥❤ ➤Ị ✶✳✶✳

tư

✭◆❣✉②➟♥ ❧ý s♦ s➳♥❤✮✳ ❈❤♦

R

❧➭ ♠ét t♦➳♥ tư tõ

B

✈➭♦

B✳

❚♦➳♥

R ➤➢ỵ❝ ❣ä✐ ❧➭ ❜➯♦ t♦➭♥ t❤ø tù ♥Õ✉ t❤á❛ ♠➲♥✿
w ⇒ R[v]

v
◆Õ✉ ❞➲②

vn

❱➭ ✈í✐ ❞➲②

✈➭ ♥Õ✉

v0

R[w]

✭✶✳✸✳✶✮

t❤á❛ ♠➲♥ ❜✃t ➤➻♥❣ t❤ø❝✿

wn

R[vn ].

✭✶✳✸✳✷✮

wn+1

R[wn ],

✭✶✳✸✳✸✮

t❤á❛ ♠➲♥✿

w0

❈❤ø♥❣ ♠✐♥❤✳

vn+1

vn

t❤×

❚õ v0

wn , ∀n ∈ N ✳
w0 s✉② r❛
R[v0 ]

R[w0 ].

▼➷t ❦❤➳❝ t❛ ❝ã

v1

R[v0 ], w1

R[w0 ]

❞♦ ➤ã

v1

R[v0 ]

R[w0 ]

w1 .

v2

R[v1 ]

R[w1 ]

w2

❚➢➡♥❣ tù✿

...............................
vn

▼Ư♥❤ ➤Ị ✶✳✷✳

R[w0 ]

❈❤♦

R[vn−1 ]

R[wn−1 ]

wn , ∀n ∈ N.

R ❧➭ ♠ét t♦➳♥ tö tõ B ✈➭♦ B✳ ❚♦➳♥ tö R t❤á❛ ♠➲♥✭✶✳✸✳✶✮✱✈➭

w0 ế wn

ị

wn+1 = R[wn ] tì wn+1

wn , ∀n ∈ N ✳


❈❤ø♥❣ ♠✐♥❤✳

❈❤ä♥

vn = wn+1 , ∀n ∈ N
t❤×

v0 = w1 = R[w0 ]

w0 ,

✈➭ ➳♣ ❞ơ♥❣ ✭✶✳✸✳✶✮ tÝ♥❤ ❝❤✃t ❝đ❛ R t❛ ❝ã✿

v1 = w2 = R[w1 ]

R[w0 ] = w1 = v0

v2 = w3 = R[w2 ]

R[w1 ] = w2 = v1

.......................................
vn = wn+1 = R[wn ]
s✉② r❛

✶✳✹

wn+1

R[wn−1 ] = wn = vn−1

wn ✱ ∀n ∈ N ✳

❳➞② ❞ù♥❣ tè❝ ➤é sã♥❣

❚r♦♥❣ ♣❤➬♥ ♥➭② t❛ sÏ ①➳❝ ➤Þ♥❤ tè❝ ➤é sã♥❣ c∗ (ξ) t➢➡♥❣ ø♥❣ ✈í✐ t♦➳♥ tư

Q

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✶✳✷✳✶✮✳ ❚è❝ ➤é sã♥❣ sÏ ➤➢ỵ❝ ➤✐Ị✉ ❝❤Ø♥❤ tr♦♥❣ ❝➳❝ ❦Õt q✉➯
➤Þ♥❤ ❧ý ë ♣❤➞♥ t✐Õ♣ t❤❡♦✳ ❚❛ sÏ ➤Þ♥❤ ♥❣❤Ü❛ c∗ (ξ) ✲ tè❝ ➤é sã♥❣✱ ✈➭ ➤➢❛ r❛ ♠ét sè
❦Õt q✉➯ ❧✐➟♥ q✉❛♥✳ ❚è❝ ➤é sã♥❣ c () ợ ị ĩ trị ớ ♥❤➢ ❧➭
❤➭♠ ❝đ❛ ➤➡♥ ✈Þ ✈❡❝t♦r ξ ✳ ❈❤ó♥❣ t❛ ❝ã t❤Ó ❤✐Ó✉ c∗ (ξ) ❧➭ tè❝ ➤é sã♥❣ t❤❡♦ ♣❤➢➡♥❣

ξ ✳ ➜Ĩ ①➳❝ ➤Þ♥❤ c∗ (ξ) t❛ ❝❤ä♥ ❤➭♠ ó tí t ợ ị ĩ t ị ĩ
◆❤÷♥❣ ❤➭♠ ❝ã tÝ♥❤ ❝❤✃t tr➟♥ s❛✉ ♥➭② t❛ ❦ý ❤✐Ö✉ ❧➭ ❤➭♠

ϕ(s)✳ ❚❛ ❝❤ä♥ ❤➭♠

ϕ(s) ♥❤➢ s❛✉✿
❍➭♠

ϕ(s) ❧➭ ♠ét ❤➭♠ sè ♥❤❐♥ ❣✐➳ trÞ t❤ù❝ t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t✿

(i) ϕ ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tô❝ ❦❤➠♥❣ t➝♥❣✳

(ii) ϕ(−∞) ∈ (π0 , π1 )✳

✭✶✳✹✳✶✮

✶✸


(iii) ϕ(s) = 0✱ ∀s
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳

0.
❱í✐ sè t❤ù❝

c

✈➭ ✈❡❝t♦r

ξ✱

t♦➳♥ tử ợ ị t

tứ

Rc, [a](s) max{(s), Q[a(c, ξ; x · ξ + s + c)](0)}.
❍❛✐ tÝ♥❤ ❝❤✃t ❝đ❛
✭✐✮ ❱í✐

✭✶✳✹✳✷✮

R s✉② r❛ tõ ➤Þ♥❤ ♥❣❤Ü❛✳

Q t❤➯♦ ♠➲♥ tÝ♥❤ ❝❤✃t ✭✶✳✷✳✶✮ ❦❤✐ ➤ã t❛ ❝ã✿

Rc,ξ [α](s) > α(c, ξ; x · ξ + s + c) = α, ✈í✐ α ∈ (π0 , π1 ).

✭✶✳✹✳✸✮

❚❤❐t ✈❐②✿

Rc,ξ [α](s) ≡ max{ϕ(s), Q[α(c, ξ; x · ξ + s + c)](0)}
Q[α(c, ξ; x · ξ + s + c)](0) = Q[α](0)
> α, t❤❡♦ tÝ♥❤ ❝❤✃t ✭✶✳✷✳✶✮ ❝đ❛ Q,
✈í✐

α ∈ (π0 , π1 ), ∀x ∈ H.
✭✐✐✮ ❱í✐

Q t❤➯♦ ♠➲♥ tÝ♥❤ ❝❤✃t ✭✐✈✮ tr ị ĩ ó ế u

tì

Rc, [u](s)

Rc, [v](s).

t ✈❐②✿
❚õ tÝ♥❤ ❝❤✃t ✭✐✈✮ ❝đ❛

Q ♥Õ✉ u

v t❤×

Q[u(c, ξ; x · ξ + s + c)](0)

Q[v(c, ξ; x · ξ + s + c)](0).

▼➷t ❦❤➳❝

Rc,ξ [u](s) ≡ max{ϕ(s), Q[u(c, ξ; x · ξ + s + c)](0)}
max{ϕ(s), Q[v(c, ξ; x · ξ + s + c)](0)}
≡ Rc,ξ [u](s).
✶✹

v




ó
ị ĩ

ế

u

v tì Rc, [u](s)

Rc,

ệ

ớ sè t❤ù❝

Rc,ξ [v](s).

c ✈➭ ✈❡❝t♦r ξ ✱ ❞➲② an (c, , s) ợ ị ở


an+1 = Rc, [an ], a0 = ϕ.
❉➢í✐ ➤➞② ❧➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❞➲②
❇ỉ ➤Ị ✶✳✶✳

tơ❝ t❤❡♦

❉➲②

an (c, ξ, s)

an ✿

❦❤➠♥❣ ❣✐➯♠ t❤❡♦

n✱

❦❤➠♥❣ t➝♥❣ t❤❡♦

s

✈➭

c✱

✈➭ ❧✐➟♥

c, ξ ✱ ✈➭ s✳

❈❤ø♥❣ ♠✐♥❤✳

✰✮ ❚❛ ❝❤ø♥❣ ♠✐♥❤ an ❦❤➠♥❣ ❣✐➯♠ t❤❡♦

n✳

❚❛ ❝ã

a0 (c, ξ; s) = ϕ(s),
s✉② r❛

a1 (c, ξ; s) = Rc,ξ [a0 ](s)
= max{ϕ(s), Q[a0 (c, ξ; x · ξ + s + c)](0)}
ϕ(s) = a0 (c, ξ; s).
❉♦

Rc,ξ ❧➭ t♦➳♥ tư ➤➡♥ ➤✐Ư✉✱ tõ ♠Ư♥❤ ➤Ị ✶✳✷ t❛ ❝ã
an+1 (c, ξ; s)

an (c, ξ; s).

❙✉② r❛ ❞➲② an (c, ξ; s) ❧➭ ❞➲② ❦❤➠♥❣ ❣✐➯♠ t❤❡♦
✰✮ ❚❛ ❝❤ø♥❣ ♠✐♥❤ an ❦❤➠♥❣ t➝♥❣ t❤❡♦
❱í✐

❝đ❛ ❤➭♠

n✳

s ✈➭ c t❤❡♦ ♣❤➢➡♥❣ ♣❤➳♣ q✉② ♥➵♣✳

n = 0 t❤× a0 (c, ξ; s) = ϕ(s) ❧➭ ♠ét ❤➭♠ ❦❤➠♥❣ t➝♥❣ t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛

ϕ(s)✱ s✉② r❛ ♥Õ✉ ∀c

c; ∀s

s t❤×

ϕ(s )

ϕ(s).
✶✺


●✐➯ sö an (c, ξ; s) ❧➭ ❦❤➠♥❣ t➝♥❣ t❤❡♦
❚ø❝ ❧➭

∀c

c; ∀s

c, s ❧➭ ➤ó♥❣✳

s t❤×

an (c , ξ; s )

an (c, ξ; s).

❚❛ ❝❤ø♥❣ ♠✐♥❤

an+1 (c , ξ; s )

an+1 (c, ξ; s), ∀c

c; ∀s

s.

❚❤❐t ✈❐②✿

∀c

c; ∀s

s s✉② r❛ x · ξ + s + c

x · ξ + s + c, ✈➭ tõ tÝ♥❤ ❝❤✃t ➤➡♥

➤✐Ư✉ ❝đ❛ an t❛ ❝ã✿

an (c , ξ; x · ξ + s + c )
♥➟♥ t❤❡♦ tÝ♥❤ ❝❤✃t ✭✐✈✮ ❝ñ❛

an (c, ξ; x · ξ + s + c),

Q t❛ ❝ã

Q[an (c , ξ; x · ξ + s + c)](0)

Q[an (c, ξ; x · ξ + s + c)](0).

❑❤✐ ➤ã

an+1 (c , ξ; s ) = Rc ,ξ [an (c , ξ; s )]
= max{ϕ(s ), Q[an (c , ξ; x · ξ + s + c )](0)}
max{ϕ(s), Q[an (c, ξ; x · ξ + s + c)](0)}
= an+1 (c, ξ; s).
◆❤➢ ✈❐② an (c, ξ; s) ❦❤➠♥❣ t➝♥❣ t❤❡♦
✰✮ ❚❛ ❝❤ø♥❣ ♠✐♥❤

s ✈➭ c ✈í✐ ♠ä✐ n✳

an (c, ξ; s) ❧✐➟♥ tô❝ t❤❡♦ ❝➳❝ ❜✐Õ♥ c, s, ξ ❜➺♥❣ ♣❤➢➡♥❣

♣❤➳♣ q✉② ♥➵♣✳
❱í✐
❤➭♠

n = 0 t❤× a0 (c, ξ; s) = ϕ(s) ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tơ❝ t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛

ϕ(s)✳ ❙✉② r❛ a0 (c, ξ; s) ❧✐➟♥ ❧ô❝ t❤❡♦ ❝➳❝ ❜✐Õ♥ c, s, ξ ✳
●✐➯ sö an (c, ξ; s) ❧✐➟♥ tô❝ t❤❡♦ ❝➳❝ ❜✐Õ♥

❚ø❝ ❧➭ ♠ä✐ ❞➲②

c, s, ξ ❧➭ ➤ó♥❣✳

(cυ , ξυ , sυ ) ❤é✐ tơ ➤Õ♥ (c, ξ; s) ❦❤✐ υ → ∞ t❤× an (cυ , ξυ ; sυ ) ❤é✐
✶✻


tô ➤Õ♥ an (c, ξ; s) ❦❤✐

υ → ∞✳

❚❛ ❝❤ø♥❣ ♠✐♥❤ an+1 (cυ , ξυ , sυ ) ❤é✐ tô ế an+1 (c, ; s)



t
ét ỗ


|x|

x t❤✉é❝ ♠ét t❐♣ ❜Þ ❝❤➷♥ B1 tr♦♥❣ H ✱ s✉② r❛ tå♥ t➵✐ sè ❞➢➡♥❣ R s❛♦
R, ∀x ∈ B1 ✳

●✐➯ sư

(cυ , ξυ , sυ ) ❤é✐ tơ ➤Õ♥ (c, ξ; s) ❦❤✐ υ → ∞✱ s✉② r❛ ❞➲② (cυ , ξυ , sυ )

❜Þ ❝❤➷♥✱ tø❝ ❧➭ ❝ã ❝➳❝ sè

M1 > 0; M2 > 0; M3 > 0 tå♥ t➵✐ υ0 > 0 s❛♦ ❝❤♦

∀υ > υ0 t❤×
|c|

M1 , |ξ|

M2 , |s|

|cυ |

M1 , |ξυ |

M2 , |sυ |

M3 ,

✈➭

M3 .

❙✉② r❛

|x · ξ + c + s|

|x|.|ξ| + |c| + |s| = M1 + RM2 + M3

✈➭

|x · ξυ + cυ + sυ |

|x|.|ξυ | + |cυ | + |sυ | = M1 + RM2 + M3 .

❑❤✐ ➤ã t❐♣

M = [−M1 ; M1 ] × [−M2 ; M2 ] × [−(M1 + RM2 + M3 ); M1 + RM2 + M3 ]
❧➭ t❐♣ ➤ã♥❣ ✈➭ ❜Þ ❝❤➷♥✱
➤✐Ĩ♠

M ❧➭ t❐♣ ❤÷✉ ❤➵♥ s✉② r❛ M ❧➭ ♠ét t❐♣ ❝♦♠♣❛❝t✱ ✈➭ ❤❛✐

(cυ , ξυ , x · ξυ + sυ + cυ )✱ (c, ξ, x · ξ + s + c) ♥➺♠ tr♦♥❣ M ✳
❱× an ❧➭ ❤➭♠ ❧✐➟♥ tô❝ t❤❡♦ t✃t ❝➯ ❝➳❝ ❜✐Õ♥ tr➟♥ t❐♣ ❝♦♠♣❛❝t M s✉② r❛ an

❧✐➟♥ tơ❝ ➤Ị✉ tr➟♥
t❤✉é❝

M ✱ ♥❣❤Ü❛ ❧➭ ∀ > 0, ∃δ > 0 s❛♦ ❝❤♦ ∀(c1 , ξ1 , s1 ), ∀(c2 , ξ2 , s2 )

M t❤á❛ ♠➲♥
|(c1 , ξ1 , s1 ) − (c2 , ξ2 , s2 )| < δ

t❤×

|an (c1 , ξ1 , s1 ) − an (c2 , ξ2 , s2 )| < .
✶✼


❱× ❞➲② (cυ , ξυ , sυ ) ❤é✐ tơ ➤Õ♥ (c, ξ; s) ❦❤✐ υ
s❛♦ ❝❤♦

→ ∞✱ tø❝ ❧➭ ∀δ > 0✱ ∃υ0 > 0

∀υ > υ0 t❤×
δ
δ
δ
|cυ − c| < , |ξυ − ξ| <
, |sυ − s| < .
3
3R
3

❙✉② r❛

|x · ξυ + cυ + sυ − (x · ξ + c + s)|

|x|.|ξυ − ξ| + |cυ − c| + |sυ − s|
< R.

◆❤➢ ✈❐②✿

δ
δ δ
+ + = δ.
3R 3 3

∀ > 0, ∃δ > 0, ∃υ0 > 0✱ ∀υ > υ0 t❤á❛ ♠➲♥
δ
δ

δ
|cυ − c| < ; |ξυ − ξ| <
; |sυ − s| < ,
9
9R
9

❦❤✐ ➤ã

∀x ∈ B1 t❛ ❝ã (cυ , ξυ , x · ξυ + sυ + cυ )✱ (c, ξ, x · ξ + s + c) tr♦♥❣ M t❤á❛

♠➲♥

|(cυ , ξυ , x · ξυ + sυ + cυ ) − (c, ξ, x · ξ + s + c)| < δ
t❤×

|an (cυ , ξυ , sυ ) − an (c, ξ, s)| < ,
❤❛②

an (cυ , ξυ ; x · ξυ + cυ + sυ ) ❤é✐ tô ➤Ò✉ ➤Õ♥ an (c, ξ; x · ξ + c + s) tr➟♥ ♠ä✐

t❐♣ ❜Þ ❝❤➷♥ tr♦♥❣

H✳

❙ư ❞ơ♥❣ tÝ♥❤ ❝❤✃t ✭✶✳✷✳✶✈✮ t❛ ❝ã

Q[an (cυ , ξυ ; x · ξυ + sυ + cυ )](0) → Q[an (c, ξ; x · ξ + c + s)](0)
❦❤✐

υ → ∞✳
▼➷t ❦❤➳❝

an+1 (cυ , ξυ , sυ ) = Rcυ ,ξυ [an (cυ , ξυ , sυ )]
= max{ϕ(sυ ), Q[an (cυ , ξυ ; x · ξυ + sυ + cυ )](0)}.

✶✽


❱×

ϕ(sυ ) → ϕ(s)
✈➭

Q[an (cυ , ξυ ; x · ξυ + sυ + cυ ](0) → Q[an (c, ξ; x · ξ + c + s)](0)
❦❤✐

υ → ∞✱ s✉② r❛
lim max{ϕ(sυ ), Q[an (cυ , ξυ ; x · ξυ + sυ + cυ )](0)}

υ→∞

= max{ϕ(s), Q[an (c, ξ; x · ξ + c + s))](0)},
❤❛②

lim an+1 (cυ , ξυ , sυ ) = an+1 (c, ξ; s).

υ→∞

❱❐② an (c, ξ; s) ❧✐➟♥ tơ❝ t❤❡♦ ❝➳❝ ❜✐Õ♥

❇ỉ ➤Ị ✶✳✷✳

c, s, ξ ✈í✐ ♠ä✐ n ∈ N ✳

❈❤♦ ❤❛✐ ❞➲② số ợ ị

n+1 = Q[n ],

n+1 = Q[n ].

α0 = ϕ(−∞),

✭✶✳✹✳✺✮

γ0 = 0.

❚❤× t❛ ❝ã ❦Õt q✉➯ s❛✉✿
✭✐✮

αn

✭✐✐✮

γn

t➝♥❣ ➤Õ♥

π1

❦❤✐

n → +∞✳

t➝♥❣ ➤Õ♥ ♠ét ♥❣❤✐Ư♠ ♥❤á ♥❤✃t ❦❤➠♥❣ ➞♠

γ

❝đ❛ ♣❤➢➡♥❣ tr×♥❤

γ = Q[γ]
❦❤✐

n → +∞✳
✭✐✐✐✮

∀c, ∀ξ ✱ ✈➭ ∀n ∈ N

❦❤✐ ➤ã

an (c, ξ; −∞) = αn

an (c, ξ; +∞) = γn .

✶✾

✭✶✳✹✳✻✮


❈❤ø♥❣ ♠✐♥❤✳

❚õ tÝ♥❤ ❝❤✃t ✭✶✳✷✳✶✐✐✐✮✱

Q[α0 ] > α0
✈➭ ✭✶✳✷✳✶✱✐✮

Q[γ0 ]
❚õ ♠Ư♥❤ ➤Ị ✶✳✷ s✉② r❛ ❞➲②

γ0 .

αn ✈➭ γn ❦❤➠♥❣ ❣✐➯♠✳

❚õ ✭✶✳✸✳✶✮ s✉② r❛

γn
❈❤♦

π0

αn

π1 .

n → +∞ ❣✐➯ sö
αn → α, γn → γ

❦❤✐ ➤ã

π0 < α

π1 ✈➭ 0

γ

π0 .

❚❛ ❝❤ø♥❣ ♠✐♥❤

α = π1 , ✈➭ γ = Q[γ].
❚❤❐t ✈❐②✿
◆Õ✉

π0

α < π1 < +∞✱ t❤❡♦ tÝ♥❤ ❝❤✃t ✭✈✐✐✮ ❝ñ❛ Q t❛ ❝ã
Q[αn ] → Q[α] > α.

❚õ

αn → α ❦❤✐ n → +∞, s✉② r❛
αn+1 = Q[αn ] → α

❦❤✐

n → +∞.
▼➷t ❦❤➳❝ ✈×

αn ❧➭ ❞➲② ❦❤➠♥❣ ❣✐➯♠ tõ ➤ã s✉② r❛
Q[α] = α.

➜✐Ị✉ ♥➭② ✈➠ ❧ý ✈× t❤❡♦ tÝ♥❤ ❝❤✃t ✭✐✈✮ ❝ñ❛ Q t❛ ❝ã

Q[α] > α.
✷✵


❱❐②

α = π1 ,
❤❛②

lim αn = π1 .

n→+∞

❈❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù s✉② r❛ γn t➝♥❣ ➤Õ♥

γ ✈➭ t❤á❛ ♠➲♥

γ = Q[γ].
❚❛ ❝❤ø♥❣ ♠✐♥❤ ✭✶✳✹✳✻✮ ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ q✉② ♥➵♣✳
❱í✐

n = 0 t❛ ❝ã
a0 (c, ξ; −∞) = ϕ(−∞) = α0 .

✭✶✳✹✳✻✮ ❧➭ ➤ó♥❣✳
●✐➯ sư ✭✶✳✹✳✻✮ ➤ó♥❣ ✈í✐ ♥✱ tø❝ ❧➭

an (c, ξ; −∞) = αn ,

t❛ ❝❤ø♥❣ ♠✐♥❤

an+1 (c, ξ; −∞) = αn+1 .
❚❤❐t ✈❐②
❚õ

an (c, ξ; −∞) = αn
❤❛②

lim an (c, ξ; s) = αn

s→−∞

tø❝ ❧➭

∀ > 0, ∃s0 > 0( ➤đ ❧í♥)✱ ∀s < −s0 t❤×
|an (c, ξ; s) − αn | < .

❳Ðt

∀x t❤✉é❝ t❐♣ ❜Þ ❝❤➷♥ tr♦♥❣ H s✉② r❛ ∃R > 0 s❛♦ ❝❤♦ |x|
|x · ξ|

|x|.|ξ| = R.
✷✶

R, ❦❤✐ ➤ã

> 0, ỗ c ố ị k0 > 0 ➤đ ❧í♥ ✭❝❤ä♥ k0 = [R + c + s0 ] + 1✮ s❛♦
❝❤♦

∀k > k0 t❛ ❝ã
x · ξ − k + c < −k0

t❤× ❦❤✐ ➤ã ❞➲②

uk (x) ≡ an (c, ξ; x · ξ − k + c)
t❤á❛ ♠➲♥

|uk (x) − αn | = |an (c, ξ; x · ξ − k + c) − αn | < .
❱❐② ❞➲②

uk (x) ≡ an (c, ξ; x · ξ − k + c)
❤é✐ tơ ➤Ị✉ tr➟♥ ♠ä✐ t❐♣ ❜Þ ❝❤➷♥ tr♦♥❣ H ❦❤✐
❚õ tÝ♥❤ ❝❤✃t ✭✶✳✷✳✶✈✮ ❝đ❛

k → +∞.

Q s✉② r❛

lim Q[an (c, ξ; x · ξ − k + c)] = Q[αn ] = αn+1 .

k→∞

❱×

αn+1

α0

ϕ ♥➟♥
lim an+1 (c, ξ; −k)] = αn+1 .

k→∞

❉♦ an+1 ❧➭ ♠ét ❤➭♠ ❦❤➠♥❣ t➝♥❣ t❤❡♦ s s✉② r❛

an+1 (c, ξ; −∞) = αn+1 .
❱❐②

an (c, ξ; −∞) = αn , ∀n, ∀c, ∀ξ.
❈❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù t❤❛② k ❜➺♥❣

−k t❛ ❝ã

an (c, ξ; +∞) = γn , ∀n, ∀c, ∀ξ.

✷✷