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Electronic Journal of Qualitative Theory of Differential Equations

2005, No.

12, 1-22;

http://www.math.u-szeged.hu/ejqtde/

✂✁☎✄✝✆✟✞✡✠☞☛✍✌✎✠✑✏✒✠☞✆✡✓✕✔✖✞✗✆✙✘✛✚✢✜✤✣✦✥ ✧✍✓✕✔✝✆✡✄✩★✪✠✬✫✮✭✯☛✱✰✲✥ ✄✝✌✳✣✦☛✱✌✴✔✝✓✱✆✵✄✝✚✲☛✱✆

✚✲☛ ✶✷✄✸✥ ✠✺✹✻✌✼✰✢✔✝✠☞✆

✽☎✾❀✿✡❁❃❂❅❄❇❆❈✿✡❁❃❉✯❊✪❁❃❂❅❆❋❉●✾■❍❑❏■▲✷▼❖◆✡P❀❆❋❄◗❊✪❁❃❘■❙❯❚✗▼❖❚✡❉●❆✦❱❲❆❋❘■✿❳✽☎✾❀✿✡❁❇❂❩❨❖❚✗❆❋❘■❍❭❬❪◆✗❆❋❚✗❆❋✾

❫✡❴❛❵❖❜❞❝❡❴❯❢❡❜❞❣❤❝❥✐❲❦❋✐✲❧♠❴❯❢❡♥❋♦✝♣✯❴✛❢❥❣rqts❋✐✝✉✸✈❈✇②①❋❣❤③❛✐✸❝❥✉④❣❤❢❥♦⑤❦⑥✐⑧⑦❃❣r❦❋❣✵⑨✎✐✸⑩✵❶❷❵⑥❵✦❸✝✉

⑨✼❹◗❺❞❻⑥✈❽❼❞❼✛❾❞❾❞❾⑥✈❽⑦❃❣r❦❋❣✵⑨✎✐✸⑩✵❶❷❵⑥❵✦❸✝✉✸✈❈❶❷⑩❤❿❞♦✸❝④❣r✐

✐✩➀➁♣✻❴❛❣r⑩➃➂➄❴❛✐✝➅❽➆②❵❖✐✸⑩r❴❛❝❥❵⑥❣➈➇❷➉➊❴❛♥⑥❜❇❜❋➋➍➌❩❝✸✈✱❵❖✐✸①❋➎➏♥⑥❜❞♥❋❝❡❴❞➇➐s⑥①❋❣❤③➊➀➁✉④❵❈❴⑥➋➑❦❋➒❛✈➓❜❞s❈❴❛♥❋❴❛❵❽➇❲s❋①❋❣➔③t➀→✉❥❵❈❴❇➋➣❦❋➒

↔✪↕❥➙❛➛➝➜❥➞❛➟➁↕❥➙✻➟➡➠❪➟➈➢❈↕⑤➤✯↕✖➤✯➠❯➥✩➦☎➠➡➧❷➨✎➥➏➠➡➧➩↕➏➫➏➫➭➠❯➥❲➯❲↕✖➥✩➲❖➙✲➳❷➵❇➸➔➺➏➞❃➜➏➢

➻✲➼✦➽✩➾✩➚✝➪➊➶➭➾

➹❅➘✲➴➁➷➊➬➱➮➄✃❃❐✸✃⑥❒❡❮✩❰✛Ï■❒✎➬Ð➘❯Ñ●❒➏➮➁➴➁➬ÐÒ❯❐✸➴➡❒✴➴➁➷t❒✴❒❥Ó❛➬➱➮➡➴➡❒❡➘tÔ➏❒ÖÕ✸×❖➮➡Õ➩ØÐÙ➊➴➁➬➱Õ➩➘t➮Ú❐✸➘tÛ⑤❒❥Ó❞➴➁❮➡❒❡Ü⑤❐✸Ø❈➮➡Õ➩ØÐÙ➊➴➁➬➱Õ➩➘t➮

×ÝÕ➩❮❷❐✪Þt❮➡➮➁➴❷Õ➩❮➡Û➊❒❡❮②➬ÐÜ➐✃➊Ù➊Ø➱➮➁➬ÐÑ●❒⑧Û❞ß✛➘❃❐✸Ü➐➬➱Ô⑤➬Ð➘tÔ❡ØÐÙt➮➁➬➱Õ➩➘àÕ➩➘✍➴➁➬ÐÜ✢❒⑤➮❑Ô✩❐✸Ø➱❒➏➮✩á✪â■ß☎Ùt➮➁➬Ð➘➊Ò✯➮➁Ù➊➬Ð➴❑❐✸ã➊Ø➱❒

Þ➊Ó❞❒➏Û❪✃⑥Õ➩➬Ð➘✛➴Ö➴➁➷t❒➏Õ➩❮➡❒❡Ü✢➮✩❰❇Ï■❒❷➮➁➴➁ÙtÛ❞ß✪➴➁➷t❒❷Ô✩❐➩➮➡❒❭ÏÚ➷t❒❡➘✯➴➁➷t❒☞❮➁➬ÐÒ➭➷❯➴✴➷❃❐✸➘tÛ❪➮➡➬➱Û➊❒☞➷❃❐➩➮✴Ô➏Õ➩➘❯Ñ●❒❥Ó✻❐➩➮

Ï■❒❡ØÐØ✦❐➩➮❀➘tÕ➩➘tÔ➏Õ➩➘❯Ñ●❒❥ÓäÑ➭❐✸ØÐÙt❒➏➮✩á

å ❁➊æ✕ç⑧▼❖❉❯✿✗è✼❆❈❘✗✿♠é➄❚✡❉●❆❈è✸❁❃è❛êÚë ♣❪ì❋s❋⑩❤✉❥❣❤③❛✐Ú❦⑥➉❃①❈❴❛♣❪❣r➎Ú❣❤①❋➎✸⑩❤s❋✉❥❣❤❜❞①❋✉✸✈●❦❋✐✝⑩❤❢❡❴✼❦❋✐✸❝④❣❤③✛❴✛❢❡❣➔③❞✐✛✈❯➎✝❜❞①➊❢❡❝❥❴❛➎✝❢❥❣r❜❞①✟✈

✐✖í❃❢❥❝❥✐✸♣✻❴❛⑩✵✉④❜❞⑩rs⑥❢❥❣r❜❞①⑥✉✸✈❇î❇í⑥✐✝❦♠ì❖❜❞❣r①➊❢✸✈⑥❢❡❣❤♣✻✐❲✉④➎➩❴❛⑩❤✐✸✉✝➋

✽✕▲ðïòñ❥▲ð❬☎ï✗ó❪ï✟◆✡✾✟ô✸❁❃❙❛õ✱ö✪❂❩❆❈è➩è✸❍❩÷➄❙❞❆⑥õ●❍❩▼❖❘■è ➂ùø❛ú❞❶❲ø❞û❇✈❈ø❛ú❞❶❷ü❞❾⑥✈⑥ø❛ú➊⑨✼ø➊û❃➋

ý

✣✦☛þ✞✵✜➄✚❪ÿ✱✓✱✌✎✞✡✄✖✚⑧☛

❭♥❋❣❤✉✲ì❈❴✛ì✦✐✝❝⑧❣r✉⑤➎✸❜❛①❋➎✸✐✝❝❥①❋✐✝❦✂✁☞❣❤❢❡♥ ❢❡♥❋✐❪✐✖í❇❣❤✉④❢❡✐✝①❋➎✸✐✯❜❛➌❭✉❥❜❛⑩rs⑥❢❥❣r❜❞①❋✉⑤❴❛①⑥❦ ✐✖í❃❢❡❝④✐✸♣✻❴❛⑩❀✉❥❜❞⑩❤s⑥❢❡❣❤❜❞①❋✉➐➌❩❜❞❝✲❴

➎✸⑩r❴❛✉❥✉✲❜❛➌❭❣❤①❋❣❤❢❥❣Ý❴❛⑩➄③✛❴❛⑩❤s❋✐❪ì❋❝❥❜❞❵⑥⑩r✐✸♣ ➌❩❜❛❝✲❣❤♣✻ì❋s⑥⑩r✉❥❣➔③❞✐ä❦⑥➉❃①❈❴❛♣❪❣r➎❪❣r①⑥➎✸⑩rs⑥✉❥❣r❜❛①❋✉✲❜❛① ❢❡❣❤♣✻✐ä✉④➎➩❴❛⑩❤✐✸✉✸➋à❧➓❜❞❝④✐

ì❋❝④✐✸➎✸❣❤✉❥✐✝⑩❤➉❞✈❈❣❤①➓⑦❇✐✸➎✖❢❡❣r❜❛①➓ø❇✈✄✁✴✐⑤➎✸❜❛①❋✉❥❣❤❦❋✐✸❝❭❢❡♥⑥✐✢➌❩❜❛⑩r⑩r❜☎✁☞❣❤①❋❿⑧ì❋❝④❜❞❵❋⑩❤✐✸♣✍➂

y

(

t

) +

p

(

t

)

y

σ

(

t

)

F

(

t, y

(

t

))

, t

J

:= [0

, b

]

T

, t

6

=

tk, k

= 1

, . . . , m,

✆✞✝✠✟

y

(

t

+

k

)

y

(

t

k

) =

Ik

(

y

(

t

k

))

, k

= 1

, . . . , m,

y

(0) =

η,

ø

✁☞♥❋✐✝❝❥✐

T

❣r✉✴❴✲❢❡❣❤♣✻✐☞✉④➎➩❴❛⑩❤✐❛✈

F

: [0

, b

]

×

✡☞☛

→ P

(

✡✌☛

)

❣r✉✎❴⑧➎✝❜❞♣✻ì❋❴❛➎✝❢Ö③✛❴❛⑩rs❋✐✝❦☎♣✪s❋⑩➔❢❡❣➱➀→③✛❴❛⑩rs⑥✐✸❦ä♣✯❴✛ì✟✈

P

(

✡☞☛

)

❣r✉➐❢❥♥❋✐þ➌❅❴❛♣✻❣❤⑩❤➉➓❜❛➌❭❴✛⑩r⑩ù①❋❜❛①❋✐✸♣❪ì⑥❢➁➉ ✉❥s⑥❵❋✉❥✐✖❢❡✉⑧❜✛➌

✡☞☛

, Ik

C

(

✡☞☛

,

✡✌☛

)

, η

✡☞☛

,

0 =

t

0

<

t

1

< ... < tm

< tm

+1

=

b,

❴❛①❋❦✕➌❩❜❞❝❭✐✸❴❛➎➏♥

k

= 1

, . . . , m

y

(

t

+

k

) = lim

h→

0

+

y

(

tk

+

h

)

❴❛①❋❦

y

(

t

k

) = lim

h→

0

y

(

tk

+

h

)

❝④✐✸ì❋❝④✐✸✉❥✐✝①➊❢✼❢❡♥❋✐②❝❥❣❤❿❞♥➊❢✎❴❛①❋❦✯⑩❤✐✝➌➈❢Ö⑩r❣❤♣✻❣➔❢❡✉ ❜❛➌

y

(

t

)

❴✛❢

t

=

tk

❣❤①♠❢❡♥❋✐✲✉❥✐✝①❋✉❥✐þ❜❛➌➄❢❡❣r♣❪✐✢✉❥➎✸❴❛⑩r✐✝✉✸✈ ❢❥♥❈❴✛❢ ❣r✉✸✈

tk

+

h

[0

, b

]

T

➌❩❜❞❝②✐➩❴✛➎➏♥

h

❣❤①à❴ ①❋✐✝❣r❿❞♥t❵❖❜❞❝❥♥❋❜❃❜❃❦✻❜❛➌

0

❴❛①⑥❦✯❣r①✯❴❛❦❋❦❋❣➔❢❡❣❤❜❞①✟✈➊❣❤➌

tk

❣❤✉✎❝④❣r❿❞♥➊❢Ö✉❥➎✸❴✛❢❥❢❥✐✸❝❥✐✝❦✟✈❇❢❥♥❋✐✸①

y

(

t

+

k

) =

y

(

tk

)

✈✍✁☞♥❋✐✸❝④✐➩❴❛✉✝✈ ✎✑✏✓✒✕✔✖✔✘✗✚✙✜✛✢✒✕✣✥✤☎✦✧✣✩★✫✪✕✬☎✭✘✮✠✒✯✔
(2)

❣❤➌

tk

❣❤✉❭⑩r✐✖➌➈❢❭✉❥➎✸❴✛❢❥❢❥✐✸❝❥✐✝❦✟✈❋❢❡♥⑥✐✸①

y

(

t

k

) =

y

(

tk

)

, σ

❣r✉☞❴⑧➌❩s⑥①❋➎✝❢❥❣r❜❞①✕❢❡♥❋❴✛❢ ✁☞❣r⑩❤⑩ ❵❖✐➐❦❋✐✝î❋①❋✐✸❦✍⑩Ý❴✛❢❥✐✸❝❭❴✛①❋❦

y

σ

(

t

) =

y

(

σ

(

t

))

.

ë ♣✻ì⑥s❋⑩r✉④❣❤③❞✐✯❦❋❣✁ ✐✸❝④✐✸①➊❢❡❣r❴❛⑩✴✐✝q❃s❋❴✛❢❡❣❤❜❞①❋✉þ♥❈❴➩③❞✐✱❵❖✐✸➎✝❜❞♣✻✐✯❣r♣❪ì✦❜❞❝❑❢➏❴❛①➊❢⑧❣r① ❝❥✐✝➎✸✐✝①t❢ä➉❛✐➩❴❛❝④✉❪❣❤① ♣✻❴✛❢❡♥❇➀

✐✸♣✻❴✛❢❥❣r➎➩❴✛⑩ ♣❪❜❇❦⑥✐✸⑩r✉✻❜❛➌⑤❝❥✐✸❴❛⑩❲ì❋❝④❜❇➎✝✐✸✉❥✉④✐✸✉✍❴❛①❋❦ ❢❥♥❋✐✝➉✂❴❛❝❥❣❤✉❥✐à❣❤①✂ì❋♥❋✐✝①❋❜❞♣❪✐✸①❈❴ ✉④❢❥s❋❦❋❣❤✐✸❦❳❣r①✂ì⑥♥t➉❃✉④❣r➎✸✉✝✈

➎➏♥❋✐✝♣✻❣❤➎➩❴❛⑩❀❢❡✐✝➎➏♥❋①❋❜❞⑩❤❜❞❿❛➉❞✈■ì✦❜❛ì❋s❋⑩Ý❴❯❢❡❣r❜❛① ❦⑥➉❃①❈❴✛♣✻❣❤➎✸✉✸✈➄❵❋❣r❜❛❢❥✐✸➎➏♥❋①⑥❜❞⑩r❜❞❿✛➉ ❴❛①❋❦ ✐✝➎✸❜❞①❋❜❛♣✻❣❤➎✸✉✸➋ ❭♥❋✐✝❝❥✐☎♥❈❴✛✉

❵❖✐✸✐✸①✂❴ ✉❥❣❤❿❞①❋❣❤î❋➎➩❴❛①➊❢✻❦❋✐✝③❛✐✸⑩r❜❛ì❋♣✻✐✝①➊❢☎❣r① ❣r♣❪ì❋s❋⑩❤✉❥✐✱❢❡♥❋✐✝❜❞❝④➉❛✈❭❣r①◗❝④✐✸➎✝✐✸①➊❢☎➉❞✐✸❴❛❝❥✉✝✈☞✐✸✉❥ì❖✐✸➎✝❣Ý❴❛⑩❤⑩❤➉ ❣❤① ❢❡♥⑥✐

❴❛❝④✐➩❴ ❜❛➌➐❣r♣❪ì❋s❋⑩❤✉❥❣➔③❞✐✕❦❋❣✁ ✐✝❝❥✐✸①➊❢❥❣Ý❴❛⑩✼✐✝qts❈❴✛❢❡❣❤❜❞①❋✉ ✁☞❣❤❢❥♥ î⑥í❇✐✝❦✂♣✻❜❛♣✻✐✝①t❢❥✉✄✂☞✉❥✐✝✐➓❢❥♥❋✐➓♣❪❜❞①❋❜❛❿❞❝❡❴❛ì⑥♥❋✉þ❜❛➌

❫✡❴❛➅t✉❥♥❋♣❪❣r➅❯❴❛①➊❢❥♥❈❴❛♣ ↕✖➟✎➞❛➸✆☎❼❛❼✞✝➝✈❈⑦⑥❴❛♣❪❜❞❣r⑩❤✐✸①❋➅✛❜ä❴❛①❋❦✍❹■✐✝❝❥✐✸✉❑❢➁➉❃s❋➅✟☎❼✹✸✠✝✡❴❛①❋❦✕❢❡♥⑥✐⑤❝④✐✝➌❩✐✝❝❥✐✸①⑥➎✸✐✸✉☞❢❡♥⑥✐✸❝❥✐✝❣r①✟➋

ë ① ❝④✐✸➎✝✐✸①➊❢⑧➉❛✐➩❴❛❝④✉⑧❦⑥➉❃①❈❴❛♣❪❣r➎ä✐✝q❃s❋❴✛❢❡❣❤❜❞①❋✉⑤❜❞① ❢❥❣r♣❪✐þ✉❥➎➩❴✛⑩r✐✸✉✲♥❈❴➩③❞✐✻❝❥✐✸➎✝✐✸❣➔③❞✐✸❦ ♣✪s❋➎➏♥ ❴❯❢❥❢❡✐✝①➊❢❡❣r❜❛①✟➋☛✡ ✐

❝❥✐✖➌❩✐✸❝❷❢❥❜✻❢❥♥❋✐✪❵❖❜❇❜❞➅t✉ ❵t➉♠⑨✴❜❞♥⑥①❋✐✸❝➐❴✛①❋❦ ❹■✐✖❢❡✐✝❝❥✉❥❜❛①☞☎➑❻⑥✈

❾✌✝➃✈✟❫✡❴❛➅t✉❥♥⑥♣✻❣❤➅✛❴✛①t❢❥♥❈❴❛♣✑↕✖➟☞➞❛➸✍☎➣❼❛ø✌✝➄❴❛①❋❦♠❢❥❜

❢❡♥⑥✐➓❝④✐✝➌❩✐✸❝④✐✸①❋➎✝✐✸✉☎➎✝❣❤❢❥✐✸❦ ❢❥♥❋✐✸❝④✐✸❣❤①✟➋ ❭♥❋✐✱❢❡❣❤♣✻✐✱✉❥➎✸❴❛⑩r✐✝✉✻➎✸❴❛⑩r➎✝s❋⑩rs⑥✉❪♥❋❴❛✉❪❢❥❝❥✐✝♣✻✐✝①❋❦❋❜❞s❋✉✻ì✦❜✛❢❡✐✸①➊❢❥❣Ý❴❛⑩✼➌❩❜❞❝

❴❛ì❋ì⑥⑩r❣r➎✸❴✛❢❡❣❤❜❞①❋✉✎❣r①✱♣✯❴❯❢❡♥❋✐✝♣✯❴✛❢❥❣r➎✸❴❛⑩✦♣❪❜❃❦❋✐✸⑩❤✉✼❜❛➌■❝❥✐➩❴✛⑩✟ì❋❝❥❜❃➎✸✐✝✉❥✉④✐✸✉ ❴❛①❋❦✍ì❋♥❋✐✸①⑥❜❞♣✻✐✝①❈❴⑥✈⑥➌❩❜❛❝❭✐✖í⑥❴❛♣❪ì❋⑩r✐➐❣❤①

ì❋♥➊➉❃✉❥❣❤➎✸✉✸✈❃➎➏♥❋✐✝♣✻❣❤➎➩❴❛⑩⑥❢❥✐✸➎➏♥❋①⑥❜❞⑩r❜❞❿✛➉❞✈➊ì❖❜❞ì❋s❋⑩r❴✛❢❡❣❤❜❞①þ❦⑥➉❃①❈❴❛♣❪❣r➎✝✉✸✈➊❵❋❣❤❜❛❢❡✐✝➎➏♥❋①❋❜❞⑩❤❜❞❿❛➉⑧❴❛①❋❦❪✐✸➎✸❜❛①❋❜❞♣❪❣r➎✸✉✝✈➊①❋✐✸s❃➀

❝❡❴✛⑩❇①❋✐✖❢ ✁✴❜❛❝❥➅t✉✸✈t✉❥❜❃➎✸❣r❴❛⑩t✉❥➎✝❣r✐✸①⑥➎✸✐✸✉✝✈➊✉❥✐✝✐✼❢❡♥❋✐✎♣✻❜❞①⑥❜❞❿❞❝❡❴✛ì❋♥❋✉➄❜✛➌❽❶❷s❋⑩❤❵❈❴❛➎➏♥þ❴❛①⑥❦✏✎ ❣❤⑩r❿❞✐✝❝✑☎➣❼✠✝➃✈➊⑨✴❜❞♥⑥①❋✐✸❝Ú❴✛①❋❦

❹■✐✖❢❡✐✝❝❥✉❥❜❛①✒☎➑❻⑥✈

❾✌✝➃✈Ú❫✡❴❛➅t✉❥♥❋♣❪❣r➅❯❴❛①➊❢❥♥❈❴❛♣ ↕✖➟✢➞❛➸✓☎❼✛ø✞✝❭❴✛①❋❦ ❢❡❜➓❢❡♥❋✐☎❝④✐✝➌❩✐✝❝❥✐✸①⑥➎✸✐✸✉✪❢❥♥❋✐✸❝④✐✸❣❤①✟➋✕✔☞✐✸➎✸✐✝①➊❢❡⑩❤➉

✎ ✐✝①❋❦❋✐✝❝❥✉❥❜❛①✖☎

❺✌✝❲❴❛①❋❦❳⑨✴✐✝①❋➎➏♥❋❜❞♥⑥❝❡❴ò↕✖➟❪➞❛➸✗☎

✈❭û❇✈✴❺✌✝❲♥❈❴➩③❛✐ ❣r①❋❣➔❢❡❣Ý❴❯❢❡✐✸❦ ❢❡♥⑥✐ ✉④❢❡s⑥❦⑥➉✂❜❛➌⑤❣❤♣✻ì❋s⑥⑩r✉❥❣➔③❞✐

❦⑥➉❃①❈❴❛♣❪❣r➎❪✐✸qts❈❴❯❢❡❣r❜❛①❋✉✪❜❞① ❢❥❣r♣❪✐✻✉④➎➩❴❛⑩❤✐✸✉✝➋ ❭♥❋✐☎î❈❝④✉④❢⑧ì❈❴❛ì❖✐✸❝✲➌❩❜❛❝⑧❣r♣❪ì❋s❋⑩❤✉❥❣➔③❞✐❪❦⑥➉❃①❈❴❛♣❪❣r➎❪❣r①❋➎✝⑩rs❋✉④❣r❜❞①❋✉

✁✴❴❛✉❲ì❋❝④❜❞ì✦❜❛✉❥✐✸❦♠❵t➉♠⑨✴✐✝⑩Ý❴❛❝④❵❋❣➝✈❖⑨✴✐✝①❋➎➏♥❋❜❞♥❋❝❥❴✕❴❛①❋❦✙✘❲s❈❴✛♥❈❴❛❵✚☎➍ú✛✝➃➋ ë ①♠❢❥♥❋❣r✉②ì❈❴❛ì❖✐✸❝✝✈✓✁✎✐✪➎✸❜❛①t❢❥❣r①ts❋✐⑤❢❡♥❋❣❤✉

✉④❢❥s❋❦⑥➉◗❵t➉◗➎✸❜❞①⑥✉❥❣r❦⑥✐✸❝❥❣❤①❋❿ ♣❪❜❞❝❥✐➓❿❞✐✸①⑥✐✸❝❡❴✛⑩❲➎✸⑩r❴❛✉❥✉④✐✸✉✕❜❛➌✢❣❤♣✻ì❋s⑥⑩r✉❥❣➔③❞✐➓❦⑥➉❃①❈❴❛♣❪❣r➎♠❣❤①❋➎✸⑩❤s❋✉❥❣❤❜❞①❋✉✻❜❞①◗❢❡❣❤♣✻✐

✉❥➎✸❴❛⑩r✐✝✉✸➋✗✡ ✐✲✉❥♥❈❴✛⑩r⑩✗ì❋❝❥❜➭③❃❣r❦❋✐✲✐✖í❇❣r✉❑❢❡✐✝①❋➎✸✐✪❝④✐✸✉④s❋⑩❤❢❥✉ ➌❩❜❞❝☞❢❡♥❋✐✲ì❋❝❥❜❛❵❋⑩r✐✝♣ ✆ ✝✠✟

ø

➋✶❭♥❋✐⑧î❈❝④✉④❢❷❜❞①❋✐⑧❝④✐✸⑩❤❣r✐✸✉

❜❞① ❢❡♥❋✐ä①❋❜❛①❋⑩r❣❤①❋✐➩❴❛❝✢❴✛⑩❤❢❡✐✝❝❥①❈❴❯❢❡❣❤③❛✐þ❜❛➌✴❫✡✐✝❝❡❴➩➉➊➀❑⑦❇➎➏♥❋❴❛s❋❦❋✐✝❝✲❢➁➉❃ì✦✐✜☎

ü✞✝ ✁☞♥❋✐✝① ❢❡♥⑥✐❪❝④❣r❿❞♥➊❢✢♥❈❴✛①❋❦ ✉④❣r❦❋✐ä❣❤✉

➎✸❜❛①t③❛✐✖í❪③✛❴❛⑩rs⑥✐✸❦✟✈❞❢❥♥❋✐☞✉❥✐✸➎✝❜❞①❋❦✯❴❛①❋❦ä❢❡♥❋✐❭❢❡♥⑥❣r❝❥❦ä❝④✐✸⑩❤➉✪❴❛⑩r✉④❜✢❜❞①❪❢❥♥❋✐☞①❋❜❞①❋⑩❤❣r①❋✐✸❴❛❝Ú❴❛⑩➔❢❡✐✝❝❥①❈❴✛❢❥❣❤③❛✐❭❜❛➌✟❫✵✐✝❝❡❴➩➉➊➀

⑦❇➎➏♥❈❴✛s❋❦❋✐✸❝ ❢➁➉❇ì❖✐⑧❵⑥s⑥❢❲s❋①⑥❦❋✐✸❝ ✁✴✐✸❴❛➅❛✐✝❝➐➎✸❜❞①❋❦⑥❣❤❢❡❣❤❜❞①❋✉☞❜❞①♠❢❥♥❋✐✲➌❩s⑥①❋➎✝❢❥❣r❜❞①❋✉

Ik

(

k

= 1

, ..., m

)

❴❛①❋❦♠❢❥♥❋✐ ♣❪❣➔í❇✐✸❦ ❿❞✐✸①⑥✐✸❝❡❴✛⑩r❣r➒✝✐✸❦ ❫✵❣rì❋✉④➎➏♥❋❣❤❢❥➒✍❴❛①❋❦✣✢✼❴✛❝❡❴✛❢❥♥❋♦✸❜❃❦❋❜❞❝❑➉✆✤➑✉❪➎✸❜❞①❋❦⑥❣❤❢❡❣❤❜❞①❋✉ä❴❛①⑥❦ ❢❡♥⑥✐✍⑩r❴❛✉④❢❪❜❞①⑥✐✍❜❛① ❢❡♥⑥✐

î⑥í❇✐✸❦ ì❖❜❞❣r①➊❢ ❢❡♥❋✐✝❜❞❝❥✐✝♣✮➌❩❜❞❝ ➎✸❜❞①➊❢❡❝❥❴❛➎✝❢❥❣r❜❞①♠♣✪s❋⑩❤❢❥❣➔➀➝③❛❴✛⑩rs❋✐✝❦➓♣✻❴❛ì❋✉❲❦⑥s❋✐⑧❢❡❜✥✢✴❜➭③❃❣❤❢❥➒þ❴❛①❋❦ ✺ ❴❛❦❋⑩❤✐✸❝✦☎

ø✌✝

✁☞♥❋✐✝①ä❢❥♥❋✐✼❝④❣r❿❞♥➊❢ù♥❋❴❛①❋❦þ✉❥❣❤❦❋✐✼❣❤✉➄①❋❜❛❢ù①❋✐✸➎✝✐✸✉❥✉❥❴❛❝❥❣❤⑩❤➉✲➎✸❜❞①➊③❛✐✖íþ③✛❴❛⑩❤s❋✐✸❦✟➋ ❭♥❋✐✼⑩r❴❛✉④❢➄✉❥✐✸➎✖❢❡❣❤❜❞①þ❣r✉■➎✸❜❞①❋➎✝✐✸❝④①❋✐✸❦

✁☞❣❤❢❥♥ ❢❥♥❋✐✻✐✩í❇❣r✉④❢❥✐✸①❋➎✝✐☎❜❛➌✴✐✖í❃❢❡❝④✐✸♣✻❴❛⑩❀✉❥❜❞⑩❤s⑥❢❡❣❤❜❞①❋✉✢❜❛➌✼❢❥♥❋✐✯❴✛❵✦❜➭③❞✐✻♣✻✐✝①➊❢❡❣r❜❛①❋✐✸❦ ì❋❝❥❜❞❵⑥⑩r✐✸♣✳❵➊➉ s❋✉④❣r①❋❿➓❴

❝❥✐✝➎✸✐✝①t❢☞î⑥í❇✐✝❦✍ì❖❜❞❣❤①t❢✴❢❡♥❋✐✝❜❞❝❥✐✝♣ ❦❋s❋✐❲❢❥❜ ✵❷♥❈❴❛❿❛✐✧☎

ú✌✝ ➌❩❜❞❝✴❢❥♥❋✐➐✉❥s⑥♣ ❜❛➌■❴þ➎✸❜❞①➊❢❥❝❡❴❛➎✖❢❡❣r❜❛①✕♣✪s❋⑩❤❢❥❣❤③✛❴❛⑩❤s❋✐✸❦

♣✻❴❛ì❪❴❛①❋❦✻❴⑤➎✝❜❞♣✻ì⑥⑩r✐✝❢❥✐✸⑩➔➉⑧➎✝❜❞①➊❢❡❣r①ts❋❜❛s❋✉✎❜❛①❋✐☞❦❋✐✝î❈①⑥✐✸❦✯❜❛①✻❜❞❝④❦❋✐✸❝④✐✸❦✻⑨✼❴❛①❈❴✛➎➏♥✯✉❥ì❋❴❛➎✸✐✝✉✸➋ ❭♥❋✐✝✉❥✐❷❝④✐✸✉④s❋⑩❤❢❥✉

➎✸❜❛♣✻ì❋⑩❤✐✸♣❪✐✸①➊❢❭❢❡♥⑥✐❲➌❩✐✯✁✷✐✩í⑥❣❤✉④❢❥✐✸①❋➎✝✐✲❝④✐✸✉❥s⑥⑩❤❢❡✉☞❦❋✐✖③❞❜❛❢❥✐✸❦✍❢❡❜ä❦⑥➉❃①❈❴❛♣❪❣r➎➐❣❤①❋➎✸⑩❤s❋✉❥❣❤❜❞①❋✉✼❜❞①✕❢❥❣r♣❪✐➐✉❥➎➩❴✛⑩r✐✸✉✝➋

★ ✩

✜■✠❭✔✖✄✸✥ ✄✝☛✱✰✢✜■✄✝✠☞✆

✡ ✐✻✁☞❣r⑩r⑩■❵❋❝❥❣❤✐✫✪❋➉♠❝❥✐✝➎➩❴❛⑩❤⑩➄✉❥❜❛♣✻✐þ❵❈❴✛✉❥❣r➎⑧❦❋✐✝î❈①⑥❣❤❢❡❣❤❜❞①❋✉➐❴✛①❋❦ ➌❅❴❛➎✝❢❥✉❲➌❩❝❥❜❞♣✤❢❡❣r♣❪✐✸✉❷✉❥➎➩❴✛⑩r✐✸✉➐➎✸❴❛⑩r➎✝s❋⑩rs⑥✉❷❢❡♥❈❴❯❢

✁✎✐ ✁☞❣❤⑩r⑩ s⑥✉❥✐✢❣r①✕❢❡♥⑥✐⑤✉④✐✸qts❋✐✝⑩➝➋

❶✂❢❡❣❤♣✻✐✎✉❥➎➩❴✛⑩r✐

T

❣❤✉Ú❴➐①❋❜❞①❋✐✝♣✻ì❇❢➁➉✪➎✸⑩❤❜❞✉❥✐✝❦ä✉❥s❋❵❋✉④✐✝❢Ö❜❛➌ ✡☞☛

.

ë ❢Ú➌❩❜❛⑩r⑩r❜☎✁☞✉➄❢❥♥❈❴✛❢ù❢❥♥❋✐✭✬④s❋♣❪ìþ❜❞ì❖✐✸❝❡❴❯❢❡❜❞❝④✉

σ, ρ

:

T

T

❦⑥✐✝î❈①❋✐✝❦♠❵➊➉

σ

(

t

) = inf

{

s

T

:

s > t

}

❴❛①❋❦

ρ

(

t

) = sup

{

s

T

:

s < t

}

✉④s❋ì❋ì❋⑩❤✐✸♣❪✐✸①➊❢❡✐✝❦♠❵➊➉

inf

:= sup

T

❴✛①❋❦

sup

:= inf

T

❴❛❝④✐ ✁✎✐✸⑩❤⑩✡❦❋✐✝î❈①⑥✐✸❦✟➋ ❭♥⑥✐⑧ì❖❜❞❣r①➊❢

t

T

❣r✉✲⑩r✐✖➌➈❢④➀➁❦⑥✐✸①❋✉④✐❛✈ù⑩❤✐✝➌➈❢④➀→✉❥➎✸❴✛❢❥❢❥✐✸❝❥✐✝❦✟✈ù❝④❣r❿❞♥➊❢④➀→❦❋✐✸①⑥✉❥✐❛✈➄❝❥❣r❿❛♥t❢❑➀➁✉④➎➩❴✛❢④❢❡✐✸❝④✐✸❦ ❣➔➌

(3)

t, σ

(

t

)

> t

❝❥✐✝✉❥ì❖✐✸➎✝❢❥❣❤③❛✐✸⑩❤➉❛➋ ë ➌

T

♥❈❴❛✉☞❴✪❝④❣r❿❞♥➊❢④➀→✉❥➎✸❴✛❢❥❢❥✐✸❝❥✐✝❦✕♣✻❣❤①❋❣r♣⑧s❋♣

m

✈❋❦❋✐✖î❈①❋✐

T

k

:=

T

− {

m

}

❜❛❢❥♥❋✐✸❝ ✁☞❣r✉❥✐✛✈ ✉④✐✝❢

T

k

=

T

.

ë

T

♥❋❴❛✉➐❴✻⑩❤✐✝➌➈❢❑➀➁✉❥➎✸❴✛❢❥❢❥✐✸❝④✐✸❦à♣✻❴❯í❇❣r♣✪s⑥♣

M

✈✟❦❋✐✖î❈①❋✐

T

k

:=

T

− {

M

}

✂ ❜❛❢❥♥❋✐✸❝ ✁☞❣r✉❥✐✛✈Ú✉❥✐✖❢

T

k

=

T

.

❭♥⑥✐☎①❋❜❛❢❡❴✛❢❡❣❤❜❞①❋✉

[0

, b

]

,

[0

, b

)

,

❴❛①⑥❦ ✉❥❜ ❜❞①✟✈ ✁☞❣❤⑩r⑩❀❦❋✐✝①❋❜❛❢❡✐✻❢❡❣r♣❪✐✯✉④➎➩❴❛⑩❤✐✸✉ ❣r①➊❢❥✐✸❝④③✛❴❛⑩❤✉

[0

, b

] =

{

t

T

:

a

t

b

}

,

✁☞♥❋✐✝❝❥✐

0

, b

T

✁☞❣❤❢❥♥

0

< ρ

(

b

)

.

❁➊÷ù❘✗❍❩õ➭❍❅▼❖❘✂✁☎✄✝✆✟✞ ↕✖➟

X

➺✖↕➓➞ ➯ ➞❛➲❖➞❃➜➏➢ ➫✡✠⑥➞❃➜➏↕☞☛✍✌✟➢❈↕➐➧✩➵❃➲ ➜✖➟❅➛➝➠❯➲

f

:

T

X

➛➈➸➈➸☞➺✖↕à➜❥➞❛➸➈➸r↕❥➙

rd

➜➏➠❯➲❈➟❅➛➈➲❈➵❽➠❯➵➊➫✏✠❖➥❡➠✒✑➭➛❅➙❃↕❥➙✯➛➈➟✎➛r➫ä➜➏➠❯➲❈➟❅➛➈➲❈➵❽➠❯➵➊➫⑧➞❛➟☞↕❥➞❃➜➏➢✍➥✖➛✔✓❯➢❃➟✝✕❑➙❃↕✖➲⑥➫➭↕✖✠❈➠❯➛➈➲❈➟✼➞❛➲❖➙❪➢⑥➞❯➫⑧➞✯➸r↕❩➧✩➟✗✕➝➫✖➛❅➙❃↕❥➙ ➸Ð➛➈➤þ➛➈➟✴➞❛➟☞↕❥➞❃➜➏➢✘✠❈➠❯➛➈➲❈➟✚✙

➥✩➛➈➟➁↕

f

Crd

(

T

) =

Crd

(

T

, X

)

.

❁➊÷ù❘✗❍❩õ➭❍❅▼❖❘✂✁☎✄✛✁✜✞ ↕✖➟

t

T

k

,

➟❩➢❈↕

➙❃↕✖➥✖➛✚✑❯➞❛➟❅➛✚✑❛↕♠➠➡➧

f

➞❛➟

t,

➙❃↕✖➲ ➠❯➟➁↕❥➙

f

(

t

)

,

➺✖↕✕➟❩➢❈↕☎➲❈➵❇➤✢✕ ➺✖↕✖➥✒✣✛✠❖➥❡➠✒✑➭➛❅➙❃↕❥➙❪➛➈➟❭↕✥✤❞➛r➫✖➟❩➫✡✦❪➛➧ù➧➩➠❯➥✲➞❛➸➈➸

ε >

0

➟➈➢❈↕✖➥❡↕þ↕✥✤❞➛r➫✖➟❩➫⑤➞✯➲ ↕✖➛✔✓❯➢⑥➺✖➠❯➥❡➢❈➠➭➠✸➙

U

➠➡➧

t

➫✖➵❽➜➏➢✕➟❩➢⑥➞❛➟

|

f

(

σ

(

t

))

f

(

s

)

f

(

t

)[

σ

(

t

)

s

]

| ≤

ε

|

σ

(

t

)

s

|

➧➩➠❯➥✲➞❛➸➈➸

s

U,

➞❛➟★✧✩✤

t

☛ ❶✷➌❩s❋①❋➎✖❢❡❣r❜❛①

F

❣r✉❭➎➩❴✛⑩r⑩r✐✝❦✱❴❛①➊❢❡❣❤❦❋✐✸❝④❣❤③✛❴✛❢❡❣➔③❞✐❲❜✛➌

f

:

T

X

ì❋❝❥❜➭③❃❣r❦❋✐✝❦

F

(

t

) =

f

(

t

)

➌❩❜❞❝❭✐➩❴✛➎➏♥

t

T

k

.

✪ ❁✬✫ ❆❋❉●❄✭✁☎✄✛✮ ✣→➛✯✦✱✰❩➧

f

➛r➫þ➜➏➠❯➲❈➟❅➛➈➲❈➵❽➠❯➵➊➫✲✙❀➟➈➢❈↕✖➲

f rd

➜➏➠❯➲❈➟❅➛➈➲❈➵❽➠❯➵➊➫☞☛ ✣→➛➈➛✯✦✳✰➈➧

f

➛r➫✲➙❃↕✖➸Ð➟➝➞✕➙❛➛✴❲↕✖➥❡↕✖➲❈➟➃➛❅➞➊➺✸➸r↕✢➞❛➟

t

➟➈➢❈↕✖➲

f

➛r➫✪➜➏➠❯➲❈➟❅➛➈➲❈➵❈➠❯➵t➫⑧➞❛➟

t

☛ ❶✷➌❩s❋①❋➎✝❢❥❣r❜❞①

p

:

T

✡✌☛

❣❤✉❭➎➩❴❛⑩❤⑩r✐✝❦ ➥➏↕✵✓❛➥❡↕➏➫➏➫✖➛✚✑❛↕✎❣❤➌

1 +

µ

(

t

)

p

(

t

)

6

= 0

➌❩❜❞❝☞❴✛⑩r⑩

t

T

,

✁☞♥❋✐✝❝❥✐

µ

(

t

) =

σ

(

t

)

t

✈ ✁☞♥❋❣❤➎➏♥ ❣r✉❷➎✸❴❛⑩r⑩❤✐✸❦✍❢❡♥❋✐✶✓❛➥④➞❛➛➈➲❈➛➈➲ ↕➏➫➏➫Ö➧✩➵❇➲ ➜✖➟❅➛➝➠❯➲❈➋ ✡ ✐✪❦❋✐✸①⑥❜❛❢❡✐þ❵➊➉

R

+

❢❡♥⑥✐ ✉❥✐✖❢⑧❜✛➌✴❢❡♥⑥✐✻❝④✐✸❿❞❝④✐✸✉❥✉④❣❤③❛✐✻➌❩s❋①❋➎✖❢❡❣❤❜❞①❋✉✸➋ ❭♥❋✐❪❿❞✐✸①❋✐✝❝❡❴❛⑩❤❣r➒✝✐✸❦ ✐✖í❇ì✦❜❛①❋✐✸①➊❢❡❣r❴❛⑩Ú➌❩s❋①❋➎✖❢❡❣r❜❛①

ep

❣r✉✢❦❋✐✝î❋①❋✐✸❦ ❴✛✉ ❢❡♥⑥✐☎s❋①❋❣❤q❃s⑥✐✯✉❥❜❛⑩rs⑥❢❥❣r❜❞① ❜❛➌❭❢❡♥❋✐✯❣r①❋❣➔❢❡❣Ý❴✛⑩➄③✛❴❛⑩rs❋✐✻ì❋❝❥❜❛❵❋⑩r✐✝♣

y

=

p

(

t

)

y, y

(0) = 1

✈ ✁☞♥❋✐✸❝④✐

p

❣r✉⑧❴ ❝❥✐✝❿❞❝❥✐✝✉❥✉④❣❤③❞✐✢➌❩s❋①❋➎✖❢❡❣❤❜❞①✟➋Ú❶❷①✍✐✖í❇ì❋⑩❤❣r➎✝❣❤❢✼➌❩❜❞❝④♣✪s❋⑩r❴✲➌❩❜❞❝

ep

(

t,

0)

❣r✉❭❿❛❣❤③❞✐✝①➓❵➊➉

ep

(

t, s

) = exp

Z

t

s

ξ

µ

(

τ

)(

p

(

τ

))∆

τ

✁☞❣❤❢❡♥

ξh

(

z

) =

(

Log(1 + hz)

h

❣❤➌

h

6

= 0

,

z

❣❤➌

h

= 0

.

✷❈❜❞❝Ú♣❪❜❞❝❥✐❭❦❋✐✖❢➏❴❛❣❤⑩r✉✝✈➊✉❥✐✸✐ ☎➑❻✌✝➝➋ ✢✴⑩r✐➩❴✛❝❥⑩❤➉❛✈

ep

(

t, s

)

①❋✐✖③❞✐✝❝Ö③✛❴❛①❋❣❤✉❥♥❋✐✝✉✸➋ ✡ ✐☞①❋❜☎✁ ❿❞❣❤③❛✐☞✉❥❜❞♣❪✐❭➌❩s❋①❋❦❈❴❛♣❪✐✸①❃➀ ❢➏❴✛⑩■ì❋❝④❜❞ì❖✐✸❝④❢❥❣r✐✝✉ ❜❛➌Ú❢❥♥❋✐⑧✐✩í❇ì✦❜❞①⑥✐✸①➊❢❡❣r❴❛⑩✗➌❩s⑥①❋➎✝❢❥❣r❜❞①✟➋ ❫✵✐✝❢

p, q

:

T

✡☞☛

❢ ✁✎❜✕❝❥✐✝❿❞❝❥✐✝✉❥✉❥❣➔③❞✐⑧➌❩s⑥①❋➎✝❢❥❣r❜❞①❋✉✝➋

✡ ✐⑤❦⑥✐✝î❈①❋✐

p

q

=

p

+

q

+

µpq,

p

:=

p

1 +

µp

,

p

q

:=

p

(

q

)

.

(4)

❚✡❁t▼❽❉❯❁✬✫ ✁☎✄✂✁☎✄✝✆✟✞ ➳❷➫➏➫✖➵❇➤✯↕✢➟➈➢⑥➞❛➟

p, q

:

T

✡☞☛

➞❛➥❡↕➐➥❡↕✵✓❛➥❡↕➏➫➏➫✖➛✚✑❛↕ù➧✩➵❃➲ ➜✖➟➃➛➝➠❯➲⑥➫✲✙❀➟➈➢❈↕✖➲♠➟➈➢❈↕Ú➧➩➠❯➸➈➸r➠

➛➈➲✬✓ä➢❈➠❯➸➔➙✡✠

✣→➛✯✦

e

0(

t, s

)

1

➞❛➲❖➙

ep

(

t, t

)

1

☛ ✣→➛➈➛✯✦

ep

(

σ

(

t

)

, s

) = (1 +

µ

(

t

)

p

(

t

))

ep

(

t, s

);

✣→➛➈➛➈➛✯✦

1

ep

(

t, s

)

=

e

⊖p

(

t, s

);

✣→➛✚✑ ✦

ep

(

t, s

)

1

ep

(

s, t

)

=

e

⊖p

(

s, t

);

✣ ✑ ✦

ep

(

t, s

)

ep

(

s, r

) =

ep

(

t, r

);

✣ ✑➭➛✯✦

ep

(

t, s

)

eq

(

t, s

) =

ep⊕q

(

t, s

);

✣ ✑➭➛➈➛✯✦

ep

(

t, s

)

eq

(

t, s

)

=

ep⊖q

(

t, s

)

.

C

([0

, b

]

,

✡✌☛

)

❣r✉✴❢❥♥❋✐⑤⑨✼❴❛①❈❴✛➎➏♥➓✉④ì❈❴❛➎✸✐✢❜❛➌➄❴❛⑩❤⑩✟➎✸❜❞①➊❢❥❣r①ts❋❜❞s❋✉✼➌❩s⑥①❋➎✝❢❥❣r❜❞①❋✉❭➌❩❝④❜❞♣

[0

, b

]

❣❤①➊❢❡❜ ✡✌☛

✁☞❣❤❢❡♥

❢❡♥⑥✐⑤①⑥❜❞❝❥♣

k

y

k

= sup

{|

y

(

t

)

|

:

t

[0

, b

]

}

.

L

1

([0

, b

]

,

✡✌☛

)

❦⑥✐✸①❋❜❛❢❥✐✻❢❥♥❋✐✯✉④ì❈❴❛➎✸✐✻❜❛➌✼➌❩s❋①⑥➎✝❢❡❣❤❜❞①❋✉✢➌❩❝❥❜❞♣

[0

, b

]

❣r①➊❢❥❜ ✡☞☛

✁☞♥⑥❣r➎➏♥ ❴❛❝④✐☎❫✵✐✸❵❖✐✸✉④❿❞s❋✐

❣r①➊❢❥✐✸❿❞❝❥❴❛❵❋⑩r✐❷❣r①✕❢❡♥❋✐➐❢❥❣r♣❪✐❲✉❥➎✸❴❛⑩r✐✢✉❥✐✝①❋✉❥✐⑤①❋❜❞❝④♣✻✐✝❦✱❵t➉

k

y

k

L

1

=

Z

b

0

|

y

(

t

)

|

t

➌❩❜❞❝❭✐✸❴❛➎➏♥

y

L

1

([0

, b

]

,

✡☞☛

)

AC

((0

, b

)

,

✡✌☛

)

❣❤✉✗❢❥♥❋✐✼✉④ì❈❴❛➎✸✐❭❜✛➌❽❦❋❣

✐✸❝④✐✸①➊❢❡❣r❴❛❵❋⑩r✐❀➌❩s❋①❋➎✖❢❡❣❤❜❞①❋✉

y

: (0

, b

)

✡☞☛

✁☞♥❋❜❛✉❥✐✼î❈❝④✉④❢Ú❦❋✐✝⑩❤❢❡❴

❦❋✐✝❝❥❣❤③✛❴✛❢❥❣❤③❛✐❛✈

y

✈❈❣❤✉☞❴❛❵❋✉④❜❞⑩rs⑥❢❥✐✸⑩➔➉☎➎✸❜❛①t❢❥❣r①ts❋❜❞s⑥✉✸➋ ❫✵✐✝❢

(

X,

| · |

)

❵❖✐☎❴➓①❋❜❞❝❥♣❪✐✸❦ ✉❥ì❈❴❛➎✝✐❛✈

P

(

X

) =

{

Y

X

:

Y

6

=

∅}

P

cl

(

X

) =

{

Y

P

(

X

) :

Y

➎✸⑩❤❜❞✉❥✐✝❦

}

P

b

(

X

) =

{

Y

∈ P

(

X

) :

Y

❵✦❜❞s⑥①❋❦❋✐✸❦

}

,

P

c

(

X

) =

{

Y

P

(

X

) :

Y

➎✸❜❛①t③❛✐✖í

}

,

P

cp

(

X

) =

{

Y

∈ P

(

X

) :

Y

➎✸❜❞♣❪ì❈❴❛➎✖❢

}

.

❶ ♣✪s⑥⑩❤❢❡❣➔③✛❴❛⑩rs❋✐✝❦ ♣✯❴❛ì

N

: [0

, b

]

P

cl

(

✡☞☛

)

❣r✉➐✉❡❴✛❣r❦ ❢❡❜✱❵❖✐✍➤✯↕❥➞❯➫✖➵❃➥④➞➊➺✸➸r↕✖✈✟❣❤➌❀➌❩❜❞❝➐✐✖③❞✐✝❝④➉

y

✡☞☛

✈✟❢❥♥❋✐þ➌❩s❋①❋➎✝❢❥❣r❜❞①

t

7−→

d

(

y, N

(

t

)) =

inf

{|

y

z

|

:

z

N

(

t

)

}

❣r✉✲♣✻✐✸❴❛✉❥s⑥❝❡❴❛❵❋⑩❤✐ ✁☞♥❋✐✝❝❥✐

d

❣r✉⑤❢❡♥❋✐✯♣✻✐✖❢❡❝❥❣❤➎✯❣❤①❋❦❋s❋➎✝✐✸❦ ❵➊➉ ❢❥♥❋✐✕⑨✼❴✛①❈❴❛➎➏♥ ✉❥ì❋❴❛➎✸✐

✡✌☛

➋ ë ① ✁☞♥❈❴✛❢☞➌❩❜❞⑩❤⑩r❜☎✁☞✉✸✈ ✁✎✐ ✁☞❣❤⑩r⑩✵❴✛✉❥✉❥s⑥♣✻✐✢❢❡♥❈❴❯❢②❢❥♥❋✐➐➌❩s❋①❋➎✖❢❡❣r❜❛①

F

: [0

, b

]

×

✡☞☛

→ P

(

✡☞☛

)

❣r✉

✢✼❴❛❝❥❴✛❢❡♥❋♦✝❜❇❦⑥❜❞❝④➉❛✈❈❣➝➋➑✐❛➋

t

F

(

t, x

)

❣r✉❭♣❪✐➩❴❛✉④s❋❝❡❴❛❵⑥⑩r✐❷➌❩❜❞❝❭✐✸❴❛➎➏♥

x

✡☞☛

❣❤❣

x

F

(

t, x

)

❣❤✉❭s❋ì❋ì❖✐✸❝☞✉❥✐✝♣✻❣❤➎✸❜❞①➊❢❥❣r①ts❋❜❞s❋✉✼➌❩❜❛❝☞❴❛⑩r♣❪❜❞✉④❢❭❴❛⑩r⑩

t

[0

, b

]

✈ ✷❈❜❞❝❭✐➩❴✛➎➏♥

y

C

([0

, b

]

,

✡✌☛

)

✈❈⑩❤✐✝❢

SF,y

❢❡♥❋✐✢✉❥✐✖❢②❜❛➌➄✉④✐✸⑩❤✐✸➎✝❢❥❣r❜❞①⑥✉②❜✛➌

F

❦⑥✐✝î❈①❋✐✝❦♠❵➊➉

SF,y

=

{

v

L

1

([0

, b

]

,

✡☞☛

) :

v

(

t

)

F

(

t, y

(

t

))

, a.e. t

[0

, b

]

}

.

❭♥❋✐❷➌❩❜❞⑩❤⑩r❜☎✁☞❣r①⑥❿⑧❫✵✐✸♣❪♣✻❴⑤❣❤✉✴➎✸❝④s❋➎✸❣r❴❛⑩❽❣r①✯❢❥♥❋✐❷ì⑥❝❥❜❃❜❛➌✵❜❛➌✡❜❞s⑥❝✴♣✻❴❛❣r①❪❝❥✐✝✉❥s❋⑩➔❢❡✉ ✁☞♥⑥✐✸①☎❢❡♥⑥✐❲♣✪s❋⑩➔❢❡❣❤③✛❴❛⑩➱➀

s❋✐✝❦➓♣✻❴❛ì✱♥❈❴❛✉☞➎✸❜❛①t③❛✐✖í✍③✛❴❛⑩rs⑥✐✸✉✸➂

(5)

❁ ✫ ✫ ❆✱✁ ✄✂✁ ✄☎✄✝✆ ✞ ☛ ✞ ↕✖➟

X

➺✖↕ ➞ ➯ ➞❛➲❖➞❃➜➏➢ ➫✡✠⑥➞❃➜➏↕☞☛ ✞ ↕✖➟

F

:

J

×

X

−→

Pcp,c

(

X

)

➺✖↕ ➞

➞❛➥④➞❛➟❩➢✠✟➏➠✸➙❃➠❯➥✩➦⑤➤þ➵❇➸Ð➟❅➛✚✑❯➞❛➸Ð➵❈↕❥➙þ➤❪➞ ✠♠➞❛➲❖➙⑧➸r↕✖➟

Γ

➺✖↕❲➞✪➸Ð➛➈➲ ↕❥➞❛➥❲➜➏➠❯➲❈➟➃➛➈➲❈➵❈➠❯➵➊➫❷➤❪➞ ✠ ✠❖➛➈➲✬✓❀➧✩➥➏➠❯➤

L

1

(

J, X

)

➟➁➠

C

(

J, X

)

✙❀➟➈➢❈↕✖➲ ➟➈➢❈↕þ➠✲✠❈↕✖➥❥➞❛➟➁➠❯➥

Γ

SF

:

C

(

J, X

)

−→

Pcp,c

(

C

(

J, X

))

,

y

7−→

SF

)(

y

) := Γ(

SF

(

y

))

➛r➫⑧➞✱➜✖➸r➠➭➫➭↕❥➙✘✓❛➥④➞ ✠❈➢à➠✲✠❈↕✖➥④➞❛➟➁➠❯➥✢➛➈➲

C

(

J, X

)

×

C

(

J, X

)

.

✁✯✄✸✆✟✞✡✠☞☛✱✌✴✠✬✏✒✠☞✆✡✓✱✔✩✞✗✆

✡ ✐ ✁☞❣r⑩r⑩✦❴❛✉④✉❥s❋♣❪✐❲➌❩❜❞❝✴❢❥♥❋✐➐❝❥✐✝♣✯❴❛❣❤①❋❦❋✐✝❝✴❜❛➌✡❢❡♥⑥❣r✉❭ì❈❴❛ì❖✐✸❝✎❢❡♥❈❴✛❢✸✈⑥➌❩❜❞❝✼✐✸❴❛➎➏♥

k

= 1

, . . . , m,

❢❡♥⑥✐➐ì✦❜❞❣❤①➊❢❡✉ ❜❛➌❈❣❤♣✻ì⑥s❋⑩r✉④✐

tk

❴❛❝④✐✎❝④❣r❿❞♥➊❢■❦❋✐✸①❋✉④✐❛➋ ë ①⑧❜❞❝❥❦❋✐✝❝✗❢❥❜❷❦❋✐✖î❈①❋✐✴❢❥♥❋✐✴✉④❜❞⑩rs❇❢❡❣r❜❛①⑤❜✛➌ ✆ ✝✠✟☞☛ ✆

ø

✈✠✁✴✐✼✉④♥❈❴❛⑩❤⑩❃➎✝❜❞①❋✉④❣r❦❋✐✝❝

❢❡♥⑥✐✢➌❩❜❛⑩r⑩r❜☎✁☞❣❤①❋❿✪✉❥ì❋❴❛➎✸✐✛➂

P C

=

{

y

: [0

, b

]

−→

✡☞☛

:

yk

C

(

Jk,

✡☞☛

)

, k

= 0

, . . . , m,

❴❛①❋❦✱❢❡♥⑥✐✸❝❥✐✢✐✩í⑥❣❤✉④❢

y

(

t

k

)

❴❛①❋❦

y

(

t

+

k

)

✁☞❣❤❢❥♥

y

(

t

k

) =

y

(

tk

)

, k

= 1

, . . . , m

}

,

✁☞♥❋❣❤➎➏♥➓❣❤✉☞❴❪⑨✴❴❛①❈❴❛➎➏♥✍✉❥ì❈❴✛➎✸✐ ✁☞❣❤❢❡♥✕❢❥♥❋✐✢①❋❜❞❝❥♣

k

y

k

P C

= max

{k

yk

k

J

k

, k

= 0

, . . . , m

}

,

✁☞♥❋✐✝❝❥✐

yk

❣❤✉ù❢❡♥❋✐☞❝❥✐✝✉④❢❥❝❥❣r➎✖❢❡❣❤❜❞①þ❜❛➌

y

❢❡❜

Jk

= (

tk, tk

+1]

[0

, b

]

, k

= 1

, . . . , m

✈❃❴❛①⑥❦

J

0

= [

t

0

, t

1]

.

❫✵✐✝❢②s❋✉☞✉④❢❡❴❛❝④❢②❵➊➉✕❦❋✐✖î❈①❋❣r①⑥❿ ✁☞♥❈❴❯❢ ✁✴✐⑤♣✻✐✸❴❛①✱❵➊➉✍❴ä✉④❜❞⑩rs❇❢❡❣r❜❛①✕❜❛➌➄ì⑥❝❥❜❞❵❋⑩❤✐✸♣

✆ ✝✠✟

ø

❁➊÷ù❘✗❍❩õ➭❍❅▼❖❘✂✮☎✄✝✆ ➳ ➧✩➵❃➲ ➜✖➟➃➛➝➠❯➲

y

P C

AC

(

J

\{

t

1

, . . . tm

}

,

✡☞☛

)

➛r➫②➫✸➞❛➛❅➙ä➟➁➠❪➺✖↕➐➞þ➫➭➠❯➸Ð➵❃➟❅➛➝➠❯➲ ➠➡➧

✣✍✌ ✦✏✎ ✣✒✑ ✦ä➛➧➐➟➈➢❈↕✖➥❡↕þ↕✥✤❞➛r➫✖➟❩➫⑤➞ ➧✩➵❇➲ ➜✖➟❅➛➝➠❯➲

v

L

1

([0

, b

]

,

✡☞☛

)

➫✖➵❈➜➏➢✱➟❩➢⑥➞❛➟

y

(

t

) +

p

(

t

)

y

σ

(

t

) =

v

(

t

)

➞ ☛❅↕☞☛✢➠❯➲

J

\{

tk

}

, k

= 1

, . . . , m,

➞❛➲❖➙✱➧➩➠❯➥ ↕❥➞❃➜➏➢

k

= 1

, . . . , m

✙✯➟➈➢❈↕þ➧✩➵❇➲ ➜✖➟❅➛➝➠❯➲

y

➫✸➞❛➟❅➛r➫ ✧✼↕➏➫ ➟➈➢❈↕ ➜➏➠❯➲❖➙❛➛➈➟❅➛➝➠❯➲

y

(

t

+

k

)

y

(

t

k

) =

Ik

(

y

(

t

k

))

,

➞❛➲❖➙✻➟➈➢❈↕⑤➛➈➲❈➟❅➛❅➞❛➸■➜➏➠❯➲❖➙❛➛➈➟➃➛➝➠❯➲

y

(0) =

η.

✡ ✐✢①❋✐✝✐✸❦✍❢❡♥❋✐➐➌❩❜❞⑩❤⑩r❜☎✁☞❣r①❋❿✪❴❛s❇í❇❣r⑩❤❣Ý❴❛❝❑➉✻❝④✐✸✉④s❋⑩❤❢

✉❥✐✸✐ ☎➣û✞✝

❁ ✫ ✫ ❆✱✮ ✄✛✁✜✞ ↕✖➟

p

:

T

✡✌☛ ➺✖↕

rd

➜➏➠❯➲❈➟❅➛➈➲❈➵❽➠❯➵➊➫✯➞❛➲❖➙♠➥❡↕✵✓❛➥❡↕➏➫➏➫✖➛✚✑❛↕☞☛✔✓❽➵ ✠ ✠❈➠➭➫➭↕

f

:

T

✡✌☛

rd

➜➏➠❯➲❈➟❅➛➈➲❈➵❽➠❯➵➊➫☞☛ ✞ ↕✖➟

t

0

T

,

➞❛➲❖➙

y

0

✡☞☛

.

✌✟➢❈↕✖➲ ✙

y

➛r➫⑧➟➈➢❈↕✪➵❃➲❈➛✖✕✸➵❽↕⑤➫➭➠❯➸Ð➵❃➟➃➛➝➠❯➲ ➠➡➧⑧➟➈➢❈↕þ➛➈➲❈➛➈➟❅➛❅➞❛➸ ✑❯➞❛➸Ð➵❈↕✖✠❖➥❡➠❛➺✸➸r↕✖➤

y

(

t

) +

p

(

t

)

y

σ

(

t

) =

f

(

t

)

, t

[0

, b

]

T

, t

6

=

tk, k

= 1

, . . . , m

ú

y

(

t

+

k

)

y

(

t

k

) =

Ik

(

y

(

t

k

))

, k

= 1

, . . . , m,

y

(0) =

y

0

,

ü

➛➧✲➞❛➲❖➙✱➠❯➲❈➸Ð➦❪➛➧

y

(

t

) =

e

⊖p

(

t,

0)

y

0

+

Z

t

0

e

⊖p

(

t, s

)

f

(

s

)∆

s

+

X

0

<t

k

<t

e

⊖p

(

t, tk

)

Ik

(

y

(

t

k

))

.

û

(6)

✝☎✟

❭♥❋✐➐➌❩s❋①⑥➎✝❢❡❣❤❜❞①

F

: [0

, b

]

×

✡☞☛

→ P

(

✡✌☛

)

❣r✉ ✢✼❴❛❝❥❴✛❢❡♥❋♦✝❜❇❦⑥❜❞❝④➉❛➋

✎❲❼

❭♥❋✐✝❝❥✐⑤✐✖í❇❣r✉❑❢☞➎✸❜❞①⑥✉④❢➏❴✛①t❢❥✉

ck

>

0

✉④s❋➎➏♥➓❢❥♥❈❴✛❢

|

Ik

(

x

)

| ≤

ck

➌❩❜❞❝❭✐➩❴✛➎➏♥

k

= 1

, . . . , m

❴✛①❋❦✍➌❩❜❛❝☞❴❛⑩r⑩

x

✡☞☛

.

✎❷ø

❭♥❋✐✝❝❥✐þ✐✩í⑥❣❤✉④❢✢❴✻➎✸❜❞①➊❢❡❣❤①ts❋❜❞s❋✉❲①⑥❜❞①❇➀➁❦⑥✐✸➎✸❝④✐➩❴❛✉④❣r①❋❿☎➌❩s⑥①❋➎✝❢❥❣r❜❞①

ψ

: [0

,

)

−→

(0

,

)

,

❴❪➌❩s❋①❋➎✖➀ ❢❥❣r❜❞①

p

L

1

([0

, b

]

,

✡☞☛

+)

❴❛①⑥❦♠❴þ➎✸❜❛①❋✉④❢❡❴❛①➊❢

M >

0

✉④s❋➎➏♥➓❢❥♥❈❴✛❢

k

F

(

t, x

)

k

P

= sup

{|

v

|

:

v

F

(

t, x

)

} ≤

p

(

t

)

ψ

(

|

x

|

)

➌❩❜❛❝❭✐➩❴❛➎➏♥

(

t, x

)

[0

, b

]

×

✡✌☛

,

❴❛①⑥❦

M

|

η

|

sup

t∈

[0

,b

]

e⊖p

(

t,

0) +

m

X

k

=1

ck

sup

t∈

[0

,b

]

e⊖p

(

t, tk

) +

sup

(

t,s

)

[0

,b

]

×

[0

,b

]

e⊖p

(

t, s

)

ψ

(

M

)

Z

b

0

p

(

s

)∆

s

>

1

.

❚✡❁t▼❽❉❯❁✬✫ ✮☎✄✛✮ ✓❽➵ ✠ ✠❈➠➭➫➭↕✢➟➈➢⑥➞❛➟Ö➢❃➦✲✠❈➠❯➟➈➢❈↕➏➫➭↕➏➫ ✣✁ ✌ ✦✏✎ ✣✁ ✑ ✦þ➢❈➠❯➸➔➙ ☛ ✌✟➢❈↕✖➲à➟➈➢❈↕⑧➛➈➤ ✠❖➵❃➸➫✖➛✚✑❛↕✲➙❛➦❯➲❖➞❛➤þ➛➝➜

➛➈➲ ➜✖➸Ð➵➊➫✖➛➝➠❯➲⑥➫✶✣✍✌ ✦✏✎ ✣✒✑ ✦⑧➢⑥➞❯➫⑧➞❛➟Ö➸r↕❥➞❯➫✖➟✼➠❯➲ ↕➐➫➭➠❯➸Ð➵❇➟❅➛➝➠❯➲ ➠❯➲

[0

, b

]

☛ ✂ ❉●▼✟▼☎✄ ✄ ✗❝❡❴❛①❋✉❑➌❩❜❞❝❥♣ ❢❥♥❋✐❪ì❋❝❥❜❞❵⑥⑩r✐✸♣

✆ ✝✠✟☞☛ ✆

ø

❣❤①t❢❥❜✍❴✍î⑥í❇✐✸❦ ì✦❜❛❣r①➊❢⑤ì❋❝❥❜❞❵⑥⑩r✐✸♣✍➋☛✢✴❜❞①❋✉❥❣❤❦❋✐✸❝✢❢❡♥⑥✐

❜❞ì❖✐✸❝❥❴✛❢❡❜❞❝

N

:

P C

−→ P

(

P C

)

❦⑥✐✝î❈①❋✐✝❦♠❵➊➉

N

(

y

) =

{

h

P C

:

h

(

t

) =

e⊖p

(

t,

0)

η

+

Z

t

0

e⊖p

(

t, s

)

v

(

s

)∆

s

+

X

0

<t

k

<t

e⊖p

(

t, tk

)

Ik

(

y

(

t

k

))

, v

SF,y

}

.

✪ ❁✬✫ ❆❋❉●❄✭✮☎✄✂✁ ✞ ➸r↕❥➞❛➥✖➸Ð➦ ✙❇➧✩➥➏➠❯➤ ✞ ↕✖➤þ➤❪➞ ✑ ☛✄ ✙Ú➟➈➢❈↕ ✧✩✤❇↕❥➙ ✠❈➠❯➛➈➲❈➟❩➫⑤➠➡➧

N

➞❛➥➏↕②➫➭➠❯➸Ð➵❃➟➃➛➝➠❯➲⑥➫❷➟➡➠ ✣✍✌ ✦✏✎ ✣✒✑ ✦ ☛ ✡ ✐⑧✉❥♥❋❴❛⑩r⑩✗✉❥♥❋❜☎✁ ❢❡♥❋❴✛❢

N

✉❡❴✛❢❥❣r✉❑î❈✐✸✉②❢❡♥❋✐þ❴✛✉❥✉❥s⑥♣✻ì⑥❢❥❣r❜❞①⑥✉ ❜❛➌Ú❢❥♥❋✐⑧①⑥❜❞①❋⑩r❣❤①❋✐➩❴✛❝ ❴✛⑩❤❢❡✐✝❝❥①❈❴❯❢❡❣❤③❛✐✲❜✛➌Ö❫✡✐✝❝❡❴➩➉➊➀ ⑦❇➎➏♥❈❴✛s❋❦❋✐✸❝☞❢➁➉❃ì✦✐✛➋ ❭♥❋✐✲ì⑥❝❥❜❃❜❛➌ ✁☞❣r⑩r⑩ ❵✦✐➐❿❞❣➔③❞✐✝①➓❣❤①✍✉④✐✝③❛✐✸❝❡❴✛⑩✗✉❑❢❡✐✝ì❋✉✸➋

ï õ➩❁❃é ✆❈ê

N

(

y

)

❣❤✉❭➎✸❜❞①➊③❞✐✩í✍➌❩❜❞❝❭✐➩❴❛➎➏♥

y

P C

➋ ë ①❋❦❋✐✝✐✸❦✟✈✡❣❤➌

h

1

, h

2

❵❖✐✸⑩r❜❛①❋❿☎❢❡❜

N

(

y

)

✈❖❢❡♥❋✐✝① ❢❡♥❋✐✝❝❥✐þ✐✖í❇❣❤✉④❢

v

1

, v

2

SF,y

✉❥s❋➎➏♥ ❢❡♥❋❴✛❢❲➌❩❜❞❝❲✐✸❴❛➎➏♥

t

[0

, b

]

✁✴✐✢♥❈❴➩③❞✐

hi

(

t

) =

e⊖p

(

t,

0)

η

+

Z

t

0

e⊖p

(

t, s

)

vi

(

s

)∆

s

+

X

0

<t

k

<t

e⊖p

(

t, tk

)

Ik

(

y

(

t

k

)) (

i

= 1

,

2)

.

❫✵✐✝❢

0

d

1

➋ ❭♥❋✐✸① ✈❽➌❩❜❛❝❭✐➩❴❛➎➏♥

t

[0

, b

]

✁✎✐✢♥❈❴➩③❞✐

(

dh

1

+ (1

d

)

h

2)(

t

) =

e⊖p

(

t,

0)

η

+

Z

t

0

e⊖p

(

t, s

)[

dv

1(

s

) + (1

d

)

v

2(

s

)]∆

s

+

X

0

<t

k

<t

e

⊖p

(

t, tk

)

Ik

(

y

(

t

k

))

.

(7)

⑦❇❣❤①❋➎✸✐

SF,y

❣r✉❭➎✝❜❞①➊③❞✐✩í

❵✦✐✝➎➩❴❛s⑥✉❥✐

F

♥❈❴❛✉❭➎✝❜❞①➊③❞✐✖í✍③✛❴❛⑩❤s❋✐✸✉

✈❋❢❡♥⑥✐✸①

dh

1

+ (1

d

)

h

2

N

(

y

)

.

ï õ➩❁❃é ✁ ➂

N

➤❪➞ ✠❈➫✲➺✖➠❯➵❇➲❖➙❃↕❥➙þ➫➭↕✖➟❩➫✢➛➈➲❈➟➁➠✯➺✖➠❯➵❃➲❖➙❃↕❥➙ä➫➭↕✖➟❩➫⑤➛➈➲

P C.

❫✵✐✝❢

Bq

=

{

y

P C

:

k

y

k

P C

q

}

❵✦✐②❴⑤❵✦❜❞s⑥①❋❦❋✐✸❦✻✉❥✐✖❢✼❣r①

P C

❴❛①❋❦

y

Bq

✈t❢❡♥⑥✐✸①✻➌❩❜❛❝Ö✐✸❴❛➎➏♥

h

N

(

y

)

✈⑥❢❡♥❋✐✝❝❥✐✢✐✖í❇❣❤✉④❢❡✉

v

SF,y

✉④s❋➎➏♥➓❢❥♥❈❴✛❢❭➌❩❜❞❝❭✐✸❴❛➎➏♥

t

[0

, b

]

h

(

t

) =

e⊖p

(

t,

0)

η

+

Z

t

0

e⊖p

(

t, s

)

v

(

s

)∆

s

+

X

0

<t

k

<t

e⊖p

(

t, tk

)

Ik

(

y

(

t

k

))

.

✷❈❝❥❜❞♣

✎❷❼

❴❛①❋❦

✎ ø

✁✴✐⑤♥❈❴➩③❞✐

|

h

(

t

)

| ≤ |

η

|

sup

t∈

[0

,b

]

e

⊖p

(

t,

0) +

sup

(

t,s

)

[0

,b

]

×

[0

,b

]

e

⊖p

(

t, s

)

Z

b

0

|

v

(

s

)

|

s

+

m

X

k

=0

e⊖p

(

t, tk

)

ck

≤ |

η

|

sup

t∈

[0

,b

]

e

⊖p

(

t,

0) +

sup

(

t,s

)

[0

,b

]

×

[0

,b

]

e

⊖p

(

t, s

)

Z

b

0

ψ

(

q

)

p

(

s

)∆

s

+

m

X

k

=0

sup

t∈

[0

,b

]

e

⊖p

(

t, tk

)

ck

≤ |

η

|

sup

t∈

[0

,b

]

e

⊖p

(

t,

0) +

sup

(

t,s

)

[0

,b

]

×

[0

,b

]

e

⊖p

(

t, s

)

ψ

(

q

)

k

p

k

L

1

+

m

X

k

=0

sup

t∈

[0

,b

]

e⊖p

(

t, tk

)

ck.

ï õ➩❁❃é ✮ ➂

N

➤❪➞ ✠❈➫✲➺✖➠❯➵❇➲❖➙❃↕❥➙þ➫➭↕✖➟❩➫✢➛➈➲❈➟➁➠✕↕ ✕✸➵❃➛➝➜➏➠❯➲❈➟➃➛➈➲❈➵❈➠❯➵t➫➐➫➭↕✖➟➈➫✪➠➡➧

P C

☛ ❫✵✐✝❢

u

1

, u

2

J, u

1

< u

2

❴❛①⑥❦

Bq

❵❖✐⑤❴þ❵✦❜❞s⑥①❋❦❋✐✸❦✍✉❥✐✖❢②❜❛➌

P C

❴✛✉②❣❤①➓⑦❃❢❡✐✝ìà❼þ❴❛①❋❦

y

Bq

➋ ✷❈❜❞❝❭✐➩❴✛➎➏♥

h

N

(

y

)

✈⑥❢❡♥⑥✐✸❝❥✐✢✐✩í⑥❣❤✉④❢❥✉

v

SF,y

✉④s❋➎➏♥➓❢❥♥❈❴✛❢☞➌❩❜❞❝❭✐✸❴❛➎➏♥

t

[0

, b

]

h

(

t

) =

e

⊖p

(

t,

0)

η

+

Z

t

0

e

⊖p

(

t, s

)

v

(

s

)∆

s

+

X

0

<t

k

<t

e

⊖p

(

t, tk

)

Ik

(

y

(

t

k

))

.

(8)

|

h

(

u

2

)

h

(

u

1

)

| ≤ |

e

⊖p

(

u

2

,

0)

e

⊖p

(

u

1

,

0)

||

η

|

+

ψ

(

q

)

k

p

k

L

1

Z

u

1

0

|

e⊖p

(

u

2

, s

)

e⊖p

(

u

1

, s

)

|

s

+

ψ

(

q

)

k

p

k

L

1

Z

u

2

u

1

e⊖p

(

u

2

, s

)∆

s

+

X

0

≤t

k

<u

1

|

e

⊖p

(

u

2

, tk

)

e

⊖p

(

u

1

, tk

)

|

ck

+

X

u

1

≤t

k

<u

2

e

⊖p

(

u

2

, tk

)

ck.

❭♥❋✐ä❝❥❣❤❿❞♥➊❢➐♥❈❴❛①⑥❦ ✉④❣r❦❋✐✪❢❥✐✸①❋❦⑥✉❲❢❡❜✱➒✸✐✝❝❥❜✍❴❛✉

u

2

u

1

0

➋⑧❶ ✉✢❴☎➎✸❜❞①⑥✉❥✐✸qts❋✐✝①❋➎✸✐✻❜❛➌✴⑦❃❢❡✐✝ì❋✉

❢❡❜✱ø

❢❡❜❛❿❞✐✝❢❥♥❋✐✸❝✲✁☞❣❤❢❡♥❪❢❡♥❋✐ ❶❷❝④➒✸✐✸⑩✁❯➀➁❶ ✉❥➎✝❜❞⑩r❣ ❭♥❋✐✝❜❞❝❥✐✝♣➓✈ ✁✴✐ ➎➩❴❛①✯➎✸❜❞①❋➎✝⑩rs❋❦⑥✐ ❢❡♥❋❴✛❢

N

:

P C

−→ P

(

P C

)

❣r✉ ➎✸❜❛♣✻ì❋⑩❤✐✝❢❥✐✸⑩❤➉✻➎✸❜❛①t❢❥❣r①ts❋❜❞s⑥✉✸➋

ï õ➩❁❃é ✁ ➂

N

➢⑥➞❯➫✲➞✍➜✖➸r➠➭➫➭↕❥➙✘✓❛➥④➞ ✠❈➢ ☛ ❫✵✐✝❢

yn

y∗, hn

N

(

yn

)

❴❛①❋❦

hn

h∗

➋ ✡ ✐✢①❋✐✝✐✸❦✍❢❡❜ä✉❥♥❋❜☎✁ ❢❥♥❈❴✛❢

h∗

N

(

y∗

)

hn

N

(

yn

)

♣✻✐✸❴❛①❋✉❭❢❡♥❋❴✛❢❭❢❡♥❋✐✝❝❥✐✢✐✖í❇❣❤✉④❢❡✉

vn

SF,y

n

✉❥s❋➎➏♥✍❢❡♥❈❴❯❢❭➌❩❜❞❝❭✐➩❴❛➎➏♥

t

[0

, b

]

hn

(

t

) =

e

⊖p

(

t,

0)

η

+

Z

t

0

e

⊖p

(

t, s

)

vn

(

s

)∆

s

+

X

0

<t

k

<t

e

⊖p

(

t, tk

)

Ik

(

yn

(

t

k

))

.

✡ ✐⑤♣⑧s❋✉④❢②✉❥♥⑥❜✠✁ ❢❡♥❈❴❯❢②❢❥♥❋✐✸❝④✐✢✐✖í❇❣r✉❑❢❡✉

h

SF,y

✉❥s❋➎➏♥✍❢❡♥❈❴✛❢❭➌❩❜❛❝❭✐➩❴❛➎➏♥

t

[0

, b

]

h∗

(

t

) =

e⊖p

(

t,

0)

η

+

Z

t

0

e⊖p

(

t, s

)

v∗

(

s

)∆

s

+

X

0

<t

k

<t

e⊖p

(

t, tk

)

Ik

(

y∗

(

t

k

))

.

✢✴⑩r✐✸❴❛❝❥⑩➔➉❞✈❋✉④❣r①❋➎✝✐

Ik, k

= 1

, . . . , m,

❴❛❝④✐➐➎✸❜❞①➊❢❥❣r①ts❋❜❞s❋✉✝✈ ✁✴✐✢♥❈❴➩③❞✐

hn

X

0

<t

k

<t

e⊖p

(

t, tk

)

Ik

(

yn

(

t

k

))

h∗

X

0

<t

k

<t

e⊖p

(

t, tk

)

Ik

(

y∗

(

t

k

))

P C

−→

0

,

❴❛✉

n

→ ∞

.

✢✴❜❞①❋✉④❣r❦❋✐✝❝❭❢❡♥❋✐✢➎✸❜❛①t❢❥❣r①ts❋❜❞s⑥✉②⑩❤❣r①❋✐✸❴❛❝✼❜❞ì❖✐✸❝❥❴✛❢❡❜❛❝

Γ :

L

1

([0

, b

]

,

✡✌☛

)

C

([0

, b

]

,

✡☞☛

)

❿❞❣➔③❞✐✸①✍❵➊➉

v

7−→

v

)(

t

) =

Z

t

0

e

⊖p

(

t, s

)

v

(

s

)

ds.

✷❈❝❥❜❞♣ ❫✵✐✸♣❪♣✻❴♠❼❇➋☞✸❇✈■❣❤❢⑤➌❩❜❞⑩r⑩❤❜✠✁☞✉➐❢❥♥❈❴✛❢

Γ

SF

❣r✉✲❴➓➎✝⑩r❜❞✉④✐✸❦ ❿❞❝❥❴❛ì❋♥ ❜❛ì✦✐✝❝❡❴✛❢❥❜❞❝✸➋➓❧➓❜❞❝④✐✸❜➭③❞✐✝❝✸✈ ✁✎✐ ♥❈❴➩③❞✐

hn

(

t

)

X

0

<t

k

<t

e

⊖p

(

t, tk

)

Ik

(

yn

(

t

k

))

Γ(

SF,y

n

)

.

(9)

⑦❇❣❤①❋➎✸✐

yn

y

,

❣❤❢✼➌❩❜❛⑩r⑩r❜☎✁☞✉✼➌❩❝④❜❞♣✒❫✵✐✸♣❪♣✻❴ä❼❇➋✌✸⑧❢❥♥❈❴✛❢❭➌❩❜❞❝❭✐✸❴❛➎➏♥

t

[0

, b

]

h

(

t

) =

e

⊖p

(

t,

0)

η

+

Z

t

0

e

⊖p

(

t, s

)

v

(

s

)∆

s

+

X

0

<t

k

<t

e

⊖p

(

t, tk

)

Ik

(

y

(

t

k

))

,

➌❩❜❞❝❭✉④❜❞♣✻✐

v

SF,v

ï õ➩❁❃é ✁✟ê ➳ ✠❖➥✩➛➝➠❯➥✖➛Ú➺✖➠❯➵❇➲❖➙❯➫✪➠❯➲♠➫➭➠❯➸Ð➵❃➟❅➛➝➠❯➲⑥➫☞☛

❫✵✐✝❢

y

❵❖✐➐✉❥s❋➎➏♥✱❢❡♥❋❴✛❢

y

λN

(

y

)

➌❩❜❞❝❭✉④❜❞♣✻✐

λ

(0

,

1)

➋ ❭♥❋✐✸①✟✈❋❢❥♥❋✐✸❝④✐✢✐✖í❇❣r✉❑❢❡✉

v

SF,y

✉④s❋➎➏♥ ❢❡♥❋❴✛❢❭➌❩❜❞❝❭✐➩❴✛➎➏♥

t

[0

, b

]

y

(

t

) =

λe⊖p

(

t,

0)

η

+

λ

Z

t

0

e⊖p

(

t, s

)

v

(

s

)∆

s

+

λ

X

0

<t

k

<t

e⊖p

(

t, tk

)

Ik

(

y

(

t

k

))

.

❭♥❋❣❤✉②❣❤♣✻ì⑥⑩r❣r✐✝✉✴❵➊➉

✎❲❼

❴❛①⑥❦

✎❷ø

❢❥♥❈❴✛❢➩✈❋➌❩❜❛❝❭✐➩❴❛➎➏♥

t

[0

, b

]

|

y

(

t

)

| ≤ |

η

|

sup

t∈

[0

,b

]

e

⊖p

(

t,

0) +

m

X

k

=1

ck

sup

t∈

[0

,b

]

e

⊖p

(

t, tk

)

+

sup

(

t,s

)

[0

,b

]

×

[0

,b

]

e⊖p

(

t, s

)

Z

b

0

p

(

s

)

ψ

(

|

y

(

s

)

|

)∆

s

≤ |

η

|

sup

t∈

[0

,b

]

e

⊖p

(

t,

0) +

m

X

k

=1

ck

sup

t∈

[0

,b

]

e

⊖p

(

t, tk

)

+

sup

(

t,s

)

[0

,b

]

×

[0

,b

]

e

⊖p

(

t, s

)

ψ

(

k

y

k

P C

)

Z

b

0

p

(

s

)∆

s.

✢✴❜❞①❋✉④✐✸qts❋✐✝①t❢❥⑩❤➉

k

y

k

P C

|

η

|

sup

t∈

[0

,b

]

e⊖p

(

t,

0) +

m

X

k

=1

ck

sup

t∈

[0

,b

]

e⊖p

(

t, tk

) +

sup

(

t,s

)

[0

,b

]

×

[0

,b

]

e⊖p

(

t, s

)

ψ

(

k

y

k

P C

)

Z

b

0

p

(

s

)∆

s

1

.

❭♥❋✐✝①♠❵➊➉

✎ ø

✈❇❢❡♥❋✐✝❝❥✐⑤✐✖í❇❣r✉❑❢❡✉

M

✉④s❋➎➏♥➓❢❥♥❈❴✛❢

k

y

k

P C

6

=

M.

❫✵✐✝❢

U

=

{

y

P C

:

k

y

k

P C

< M

}

.

❭♥❋✐☎❜❛ì✦✐✝❝❡❴✛❢❥❜❞❝

N

:

U

→ P

(

P C

)

❣❤✉✲s⑥ì❋ì✦✐✝❝⑧✉④✐✸♣❪❣r➎✸❜❛①t❢❥❣r①ts❋❜❞s⑥✉✪❴❛①❋❦ ➎✸❜❞♣❪ì❋⑩r✐✖❢❡✐✝⑩❤➉ ➎✝❜❞①➊❢❡❣r①ts❋❜❛s❋✉✸➋

✷❈❝❥❜❞♣ ❢❡♥⑥✐✪➎➏♥❋❜❞❣❤➎✸✐✪❜❛➌

U

✈ ❢❥♥❋✐✸❝④✐þ❣r✉ ①❋❜

y

∂U

✉❥s❋➎➏♥ ❢❥♥❈❴✛❢

y

λN

(

y

)

➌❩❜❛❝❷✉❥❜❛♣✻✐

λ

(0

,

1)

.

❶❷✉❷❴✻➎✝❜❞①❋✉❥✐✝qts❋✐✸①❋➎✝✐þ❜❛➌➄❢❥♥❋✐⑧①⑥❜❞①❋⑩r❣❤①❋✐➩❴✛❝☞❴❛⑩❤❢❥✐✸❝❥①❋❴✛❢❡❣➔③❞✐⑤❜❛➌Ú❫✵✐✸❝❥❴➩➉t➀➡⑦❇➎➏♥❈❴❛s⑥❦❋✐✸❝❷❢➁➉❃ì❖✐ ☎

ü✌✝➃✈ ✁✴✐✲❦❋✐✸❦❋s⑥➎✸✐

❢❡♥❋❴✛❢

N

♥❈❴❛✉②❴þî⑥í❇✐✸❦✍ì✦❜❞❣❤①➊❢

y

❣r①

U

✁☞♥⑥❣r➎➏♥♠❣❤✉☞❴þ✉❥❜❞⑩❤s⑥❢❡❣❤❜❞①✕❜❛➌✗❢❡♥❋✐✢ì❋❝④❜❞❵❋⑩r✐✝♣

✆✞✝✠✟☎☛ ✆

ø

✡ ✐➐①⑥❜✠✁ ì❋❝④✐✸✉❥✐✝①➊❢ ❢ ✁✎❜ä❜❛❢❡♥❋✐✝❝✼✐✖í❇❣❤✉④❢❡✐✝①❋➎✸✐⑤❝❥✐✝✉❥s❋⑩➔❢❡✉✴➌❩❜❛❝✴❢❡♥❋✐➐ì⑥❝❥❜❞❵❋⑩❤✐✸♣

✆ ✝✠✟☞☛ ✆

ø

✁☞♥❋✐✸①✱❢❡♥⑥✐➐❝❥❣r❿❛♥t❢

♥❈❴❛①⑥❦✕✉❥❣r❦⑥✐❲♥❈❴❛✉✼➎✝❜❞①➊③❞✐✩í✕③✛❴❛⑩rs❋✐✝✉✼s❋①❋❦⑥✐✸❝ ✁✎✐➩❴❛➅✛✐✸❝❭➎✸❜❛①❋❦❋❣❤❢❥❣r❜❞①⑥✉✴❜❞①☎❢❥♥❋✐❷➌❩s❋①❋➎✖❢❡❣❤❜❞①❋✉

Ik

(

k

= 1

, ..., m

)

❴✛✉②s⑥✉❥✐✸❦✍❣r①✟☎ ✝✹✝

✝✟➌❩❜❛❝❭❣r♣❪ì❋s❋⑩r✉④❣❤③❛✐❲❦❋❣✁ ✐✸❝④✐✸①➊❢❡❣r❴❛⑩✟❣❤①❋➎✸⑩❤s❋✉❥❣❤❜❞①❋✉

(10)

✣✁ ✆ ✦ ✌✟➢❈↕✖➥❡↕✪↕✥✤❞➛r➫✖➟❭➜➏➠❯➲⑥➫✖➟→➞❛➲❈➟➈➫

ck

>

0

➫✖➵❽➜➏➢✍➟➈➢⑥➞❛➟

|

Ik

(

x

)

| ≤

ck

|

x

|

➧➩➠❯➥✪↕❥➞❃➜➏➢

k

= 1

, ..., m

➞❛➲❖➙☎➞❛➸➈➸

x

✡☞☛

.

✣✁✁ ✦

Hd

(

F

(

t, y

)

, F

(

t, y

))

l

(

t

)

|

y

y

|

➧➩➠❯➥☎↕❥➞❃➜➏➢

t

[0

, b

]

➞❛➲❖➙ ➞❛➸➈➸

y, y

✡☞☛ ✎ ➢❈↕✖➥❡↕

l

L

1

([0

, b

]

,

✡☞☛

+)

∩ R

+

➞❛➲❖➙

d

(0

, F

(

t,

0))

l

(

t

)

➞ ☛❅↕

t

[0

, b

]

.

✰❩➧

sup

(

t,s

)

[0

,b

]

×

[0

,b

]

e⊖p

(

t, s

)

k

l

k

L

1

+

m

X

k

=1

sup

t∈

[0

,b

]

e⊖p

(

t, tk

)

ck

<

1

,

➟➈➢❈↕✖➲ ➟❩➢❈↕✖✠❖➥❡➠❛➺✸➸r↕✖➤ ✣✍✌ ✦✒✕ ✣✒✑ ✦⑧➢⑥➞❯➫✲➞❛➟✎➸r↕❥➞❯➫✖➟❭➠❯➲ ↕❲➫➭➠❯➸Ð➵❃➟➃➛➝➠❯➲ ➠❯➲

[0

, b

]

☛ ✂ ❉●▼✟▼☎✄ ✄ ❫✵✐✝❢

y

❵✦✐➓✉❥s❋➎➏♥✂❢❥♥❈❴✛❢

y

λN

(

y

)

➌❩❜❞❝❪✉❥❜❞♣❪✐

λ

(0

,

1)

➋ ❭♥❋✐✝①✟✈✼❢❡♥⑥✐✸❝❥✐♠✐✖í❇❣r✉❑❢

v

SF,y

✉④s❋➎➏♥➓❢❥♥❈❴✛❢❭➌❩❜❞❝❭✐✸❴❛➎➏♥

t

[0

, b

]

y

(

t

) =

λe⊖p

(

t,

0)

η

+

λ

Z

t

0

e⊖p

(

t, s

)

v

(

s

)∆

s

+

λ

X

0

<t

k

<t

e⊖p

(

t, tk

)

Ik

(

y

(

t

k

))

.

❭♥❋❣❤✉②❣❤♣✻ì⑥⑩r❣r✐✝✉✴❵➊➉

✎ ú

❴❛①⑥❦

✎ ✸

❢❥♥❈❴✛❢☞➌❩❜❞❝❭✐➩❴✛➎➏♥

t

[0

, b

]

|

y

(

t

)

| ≤ |

η

|

sup

t∈

[0

,b

]

e

⊖p

(

t,

0) +

m

X

k

=1

sup

t∈

[0

,b

]

e

⊖p

(

t, tk

)

ck

|

y

(

t

k

)

|

+

sup

(

t,s

)

[0

,b

]

×

[0

,b

]

e⊖p

(

t, s

)

Z

b

0

|

v

(

s

)

|

s.

≤ |

η

|

sup

t∈

[0

,b

]

e

⊖p

(

t,

0) +

m

X

k

=1

sup

t∈

[0

,b

]

e

⊖p

(

t, tk

)

ck

|

y

(

t

k

)

|

+

sup

(

t,s

)

[0

,b

]

×

[0

,b

]

e

⊖p

(

t, s

)

Z

b

0

|

l

(

s

)

y

(

s

) +

l

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