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,
✆❼
✟
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❣❤♣✻❣➔❢❡✉ ❜❛➌y
(
t
)
❴✛❢t
=
tk
❣❤①♠❢❡♥❋✐✲✉❥✐✝①❋✉❥✐þ❜❛➌➄❢❡❣r♣❪✐✢✉❥➎✸❴❛⑩r✐✝✉✸✈ ❢❥♥❈❴✛❢ ❣r✉✸✈tk
+
h
∈
[0
, b
]
∩
T
➌❩❜❞❝②✐➩❴✛➎➏♥h
❣❤①à❴ ①❋✐✝❣r❿❞♥t❵❖❜❞❝❥♥❋❜❃❜❃❦✻❜❛➌0
❴❛①⑥❦✯❣r①✯❴❛❦❋❦❋❣➔❢❡❣❤❜❞①✟✈➊❣❤➌tk
❣❤✉✎❝④❣r❿❞♥➊❢Ö✉❥➎✸❴✛❢❥❢❥✐✸❝❥✐✝❦✟✈❇❢❥♥❋✐✸①y
(
t
+
k
) =
y
(
tk
)
✈✍✁☞♥❋✐✸❝④✐➩❴❛✉✝✈ ✎✑✏✓✒✕✔✖✔✘✗✚✙✜✛✢✒✕✣✥✤☎✦✧✣✩★✫✪✕✬☎✭✘✮✠✒✯✔❣❤➌
tk
❣❤✉❭⑩r✐✖➌➈❢❭✉❥➎✸❴✛❢❥❢❥✐✸❝❥✐✝❦✟✈❋❢❡♥⑥✐✸①y
(
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✐✝♣ ✆ ✝✠✟
➀
✆
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❋✐✝❦♠❵➊➉σ
(
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
t, σ
(
t
)
> t
❝❥✐✝✉❥ì❖✐✸➎✝❢❥❣❤③❛✐✸⑩❤➉❛➋ ë ➌T
♥❈❴❛✉☞❴✪❝④❣r❿❞♥➊❢④➀→✉❥➎✸❴✛❢❥❢❥✐✸❝❥✐✝❦✕♣✻❣❤①❋❣r♣⑧s❋♣m
✈❋❦❋✐✖î❈①❋✐T
k
:=
T
− {
m
}
✂
❜❛❢❥♥❋✐✸❝ ✁☞❣r✉❥✐✛✈ ✉④✐✝❢
T
k
=
T
.
ë
➌
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❜❛①F
❣r✉❭➎➩❴✛⑩r⑩r✐✝❦✱❴❛①➊❢❡❣❤❦❋✐✸❝④❣❤③✛❴✛❢❡❣➔③❞✐❲❜✛➌f
:
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❜❞①p
:
T
→
✡✌☛❣❤✉❭➎➩❴❛⑩❤⑩r✐✝❦ ➥➏↕✵✓❛➥❡↕➏➫➏➫✖➛✚✑❛↕✎❣❤➌
1 +
µ
(
t
)
p
(
t
)
6
= 0
➌❩❜❞❝☞❴✛⑩r⑩t
∈
T
,
✁☞♥❋✐✝❝❥✐
µ
(
t
) =
σ
(
t
)
−
t
✈ ✁☞♥❋❣❤➎➏♥ ❣r✉❷➎✸❴❛⑩r⑩❤✐✸❦✍❢❡♥❋✐✶✓❛➥④➞❛➛➈➲❈➛➈➲ ↕➏➫➏➫Ö➧✩➵❇➲ ➜✖➟❅➛➝➠❯➲❈➋ ✡ ✐✪❦❋✐✸①⑥❜❛❢❡✐þ❵➊➉R
+
❢❡♥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❜❛①ep
❣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❵➊➉ep
(
t, s
) = exp
Z
t
s
ξ
µ
(
τ
)(
p
(
τ
))∆
τ
✁☞❣❤❢❡♥
ξh
(
z
) =
(
Log(1 + hz)
h
❣❤➌
h
6
= 0
,
z
❣❤➌h
= 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
)
.
❚✡❁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❋✐✝❦ ♣✯❴❛ìN
: [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❜✛➌F
❦⑥✐✝î❈①❋✐✝❦♠❵➊➉SF,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
❁ ✫ ✫ ❆✱✁ ✄✂✁ ✄☎✄✝✆ ✞ ☛ ✞ ↕✖➟
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−→
(Γ
◦
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❜❛➌y
❢❡❜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❤✐✸♣✆ ✝✠✟
➀
✆
ø
✟
➋
❁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✞✝
✟
➋
❁ ✫ ✫ ❆✱✮ ✄✛✁✜✞ ↕✖➟
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
✟
✆
✎
✝☎✟
❭♥❋✐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❩❜❛❝☞❴❛⑩r⑩x
∈
✡☞☛.
✆
✎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❜❞①p
∈
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❋✐✝❦♠❵➊➉N
(
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❋✉✸➋ï õ➩❁❃é ✆❈ê
N
(
y
)
❣❤✉❭➎✸❜❞①➊③❞✐✩í✍➌❩❜❞❝❭✐➩❴❛➎➏♥y
∈
P C
➋ ë ①❋❦❋✐✝✐✸❦✟✈✡❣❤➌h
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❇❣❤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❜❛➌P 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
))
.
|
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✉✸➋ï õ➩❁❃é ✁ ➂
N
➢⑥➞❯➫✲➞✍➜✖➸r➠➭➫➭↕❥➙✘✓❛➥④➞ ✠❈➢ ☛ ❫✵✐✝❢yn
→
y∗, hn
∈
N
(
yn
)
❴❛①❋❦hn
→
h∗
➋ ✡ ✐✢①❋✐✝✐✸❦✍❢❡❜ä✉❥♥❋❜☎✁ ❢❥♥❈❴✛❢h∗
∈
N
(
y∗
)
➋hn
∈
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✍❵➊➉v
7−→
(Γ
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
)
.
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
∗
➋
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
❣r①U
✁☞♥⑥❣r➎➏♥♠❣❤✉☞❴þ✉❥❜❞⑩❤s⑥❢❡❣❤❜❞①✕❜❛➌✗❢❡♥❋✐✢ì❋❝④❜❞❵❋⑩r✐✝♣✆✞✝✠✟☎☛ ✆
ø
✟
➋
✡ ✐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❋✉
✟
➋
✎
✣✁ ✆ ✦ ✌✟➢❈↕✖➥❡↕✪↕✥✤❞➛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