✱ ✁☎☞✲✞✙✘☛✠ ✳✴✄✵✠✛✁☎✄✷✶✸✆✰✹✺✁☎✄✩✁☎✘✙✍✑✆✻✔✣✼✂✽✿✾❀✏✕✔✖✍✗✆✟✞✙✘✛✚
❁❃❂❅❄❇❆❉❈✎❊●❋■❍❑❏✝❈▲❊✬▼❖◆✩❏P❆
◗☎❘❚❙❱❯❳❲✟❨❬❩❭❙❱❪❫❲❵❴✝❛❜❨❬❙❱❩❞❝❢❡☎❣❳❨❭❤
✐❢❘❥❩❞❪❫❙❱❪❬❦❚❪☎❧❵❛❜♠♥❙❱♠♥❲♦❲q♣❚r✩st❧✖s✥✉✲❡✇✈☎①✫◗✇②✒✈④③❳⑤❜⑥❜⑦
⑦❳⑥q⑧⑨⑥❳⑩❑❶❷❙❱♠♥♠♥❲❸❪❹❛❜❘❚❲♦❦❥❩❭❲❜♣❥❺❉❨❬❛❜❘❥❻♦❲
❼❞❽❿❾✌➀⑨➁➃➂q❽➅➄✻➆➈➇■➄♦➉❢❽➋➊➀⑨➌⑨➁➃➍q➎➐➏⑨➑❢➁➓➒✰➔✰→❳➊➣↔➑
↕■➙✎➛❹➜✰➝♦➞⑨➟➠➜
➡
❾✌➌❿➒❢➁➓➢⑨➤❫➑■➉❢❽❿➤❇➥❉➁➓➤❫➌➅➄♦➑➦➢❃➒❢➧❳➒❢➉➦➤❫➇
u
′′
+
f
(
u
)
u
′
+
g
(
u
) = 0
➨
➁➃➉❢❽➩➄♦➌
➁➓➒➦❾✌➫➭➄♦➉➦➤❹➢➯➏❥➤❫➑❢➁➓❾q➢❳➁➓❼✝➒➦❾✌➫➃➀⑨➉❢➁➓❾✌➌➋➊▲➲P❽⑨➁➓➒➳➏➅➄♦➏❚➤❫➑▲❼❹❾✌➌❿❼❹➤❫➑❢➌❿➒➳➉❢❽❿➤✵➵❚➤❫❽➅➄✰➍q➁➓❾✌➑✎❾♦➣❜➏❥➤❫➑❢➁➓❾q➢❳➁➓❼
➒➦❾✌➫➃➀⑨➉❢➁➓❾✌➌❿➒✑❾♦➣✩➥❉➁➓➤❫➌➅➄♦➑➦➢✙➒❢➧❳➒❢➉➦➤❫➇④➀⑨➌❿➢⑨➤❫➑❅➒➦➇■➄♦➫➃➫➋➏❚➤❫➑❢➁➓❾❜➢❳➁➓❼✲➏❥➤❫➑❢➉❢➀⑨➑❢➵➅➄♦➉❢➁➓❾✌➌❿➒✰➊
➸✇➺❫➻➽➼✇➾✌➚❢➪♦➶✰➹
➏❚➤❫➑❢➉❢➀⑨➑❢➵❥➤❹➢❷➒❢➧❳➒❢➉➦➤❫➇❵➒✰➘✌➥❉➁➓➤❫➌➅➄♦➑➦➢❷➤❹➴➷➀➅➄♦➉❢➁➓❾✌➌➋➘✌➏❥❾✌➫➃➧❜➌❿❾✌➇✖➁➭➄♦➫❳➒➦➧❜➒❢➉➦➤❫➇❵➒❸➊
➬❜➮✻➮✻➮✖➱❐✃➠❒♥❮ ➺❫❰
✃✻❒↔Ï↔Ð➦✃➠Ñ➋Ò➅Ó❿Ô♥Õ
➺
Ð❹❒➯Ö✎Ñ×✃ ➶❞➶
ÏØ✝Ð➦✃✻❒↔Ï ➾✌Ù❐Ú
→♦Û ➡PÜ➠Ý
➘➅→♦Û
➡
→
Ý
Þ ß▲à✯á✩â✝ã❇äæåæç➈á●è❸ã✣à
✉é❣❳❘❚❩❬❙♥❤❚❲♦❨➯❪❫ê❥❲❵❩❭❲♦❻♦❣❜❘❥❤✛❣❜❨❬❤❥❲✟❨t❤❚❙❱ë▲❲♦❨❬❲✟❘⑨❪❫❙↔❛q♠✮❲♦ì❿❦❉❛q❪❬❙♥❣❳❘æ❣❜í✵❪❢î➅ï▲❲
(
Eǫ
)
x
′′
+
g
(
t, x, x
′
, ǫ
) = 0
ð
ê❥❲✟❨❬❲
ǫ >
0
❙❱❩✡❛ñ❩❬ò❇❛❜♠♥♠óï❉❛q❨❫❛❜ò✙❲✟❪❬❲♦❨♦♣g
❙♥❩❀❛T
❝ôï▲❲✟❨❬❙♥❣➅❤❥❙❱❻✫í➐❦❥❘❚❻✟❪❫❙❱❣❳❘✂❙❱❘t
❛q❘❥❤g
(
t, x,
0
,
0) =
g
(
x
)
❙♥❩➯❙❱❘❥❤❥❲♦ï➳❲♦❘❚❤❥❲♦❘⑨❪t❣❜ít
õ ✐❢❘❀❪❬ê❥❲✖❻✌❛❜❩❭❲ð
ê❥❲♦❨❭❲
g
❙❱❩✲❙♥❘❚❤❥❲♦ï➳❲♦❘❥❤❚❲♦❘⑨❪ó❣❜íx
′
❛❜❘❥❤✡❙❱❩✲❻♦❣❳❘⑨❪❬❙♥❘❿❦❥❣❳❦❥❩❭♠❱î❐❤❥❙❱ë▲❲♦❨❭❲♦❘⑨❪❫❙♥❛❜ö❥♠♥❲t❙♥❘x
❪❬ê❥❲■❲❸÷ø❙♥❩❞❪❫❲♦❘❚❻♦❲❑ï❥❨❭❣❳ö❥♠❱❲♦ò✦❣❜íP❘❥❣❳❘✒❻✟❣❳❘❥❩❞❪❹❛❜❘⑨❪❷ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻✖❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘❥❩ó❣❜í(
Eǫ
)
ê❉❛q❩☎ö➳❲♦❲✟❘ ❩❞❪❫❦❥❤❥❙❱❲♦❤✫ö⑨î✡ò❐❛q❘❿î✡❛❜❦❚❪❫ê❚❣❳❨❬❩õ
✐❢❘❥❤❚❲♦❲♦❤✎♣✩❙♥❘✛❪❬ê❥❲❑♠♥❛q❪❬❪❬❲♦❨✇❻♦❛❜❩❬❲✣❻♦❲♦❨❞❪❹❛❜❙❱❘ù❛❜ò✙❣❳❘❥ú✙❪❫ê❥❲✟ò✸ï❥❨❭❣✻❯❜❲♦❤✕❲❸÷ø❙❱❩❭❪❫❲✟❘❥❻♦❲✯❣qí✑❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘❥❩
❣❜í
x
′′
+
g
(
t, x, ǫ
) = 0
.
❺❉❣❳❨➯❛➽❨❬❲✟❯➅❙❱❲ð
❩❭❲♦❲ ✉éê❥❣
ð
❝✁☎❛❜♠♥❲✄✂✉❅❝✁✆☎✵❛❜❘❥❤✝✇❛❜♠❱❲✞✂✟✆☎
õ
✠é❦ø❪❐ò❇❛❜❘⑨î✺❲❸÷❚❛❜ò✙ï❥♠❱❲♦❩☛✡ ❛❜❩➽❪❫ê❥❲✫❣❳❘❥❲æú❳❙➭❯❳❲✟❘ ö⑨î☞✇❛❜❨❞❪❫ò❇❛❜❘✍✌✣ï❥❨❭❣✻❯❜❲♦❤✂❘❚❣❳❘ ❲✰÷❚❙❱❩❭❪❬❲♦❘❥❻✟❲
❻♦❛❜❩❬❲✟❩❇❣❜í☎❪❫ê❉❛➷❪➽❲♦ì❿❦❉❛q❪❬❙♥❣❳❘ ❙❱í
ð
❲✫❩❬❦❚ï❥ï▲❣❜❩❬❲
g
❤❥❲✟ï▲❲✟❘❥❤❥❲♦❘⑨❪❐❣❳❘x
′
õ✏✎
❲✡❪❫ê❥❲✟❘ ❻♦❛❜❘❥❘❥❣❜❪
❲✰÷❚ï➳❲♦❻❸❪☎❪❬❣➽ú❳❲♦❘❥❲✟❨❫❛❜♠❱❙✒✑✟❲❷❪❫ê❥❲✟❙♥❨➯❨❬❲✟❩❬❦❥♠➭❪❫❩
õ
r✩❲✟❪ó❪❫ê❚❲❵í➐❣❜♠♥♠♥❣
ð
❙❱❘❥ú❑❲♦ì❿❦❉❛➷❪❫❙♥❣❜❘✮♣
ð
ê❥❙❱❻❹ê✛❙❱❩ó❛➽ï➳❲♦❨❞❪❫❦❥❨❭ö▲❲✟❤✒r✩❙❱❲♦❘❉❛❜❨❭❤✡❪❢îøï➳❲
(1
ǫ
)
x
′′
+
f
(
x
)
x
′
+
g
(
x
) =
ǫh
(
t
T
, x, x
′
, ǫ
)
ð
ê❥❲✟❨❬❲
h
❙♥❩T
❝❢ï➳❲♦❨❬❙❱❣ø❤❚❙♥❻❵❙♥❘t
♣f
❛❜❘❚❤g
❛❜❨❬❲■í➐❦❥❘❥❻✟❪❬❙♥❣❳❘❚❩✇❣❳❘❥♠➭î✫❤❥❲♦ï➳❲♦❘❥❤❚❲♦❘⑨❪❵❣❳❘x
♣✮❩❬❛q❪❭❝ ❙❱❩❭íî➅❙♥❘❥ú✫❻♦❣❜❘❥❤❥❙❱❪❬❙♥❣❳❘❚❩■❤❚❲✔✓❉❘❥❲✟❤ñö➳❲♦♠♥❣ð
õ✕✎
❲❇♠❱❣ø❣✗✖✒í➐❣❳❨■ï▲❲✟❨❬❙♥❣➅❤❥❙❱❻✯❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘❥❩❵❣❜í✘✡
1
ǫ
✌❷í➐❣❳❨
ǫ
❩❬ò❇❛❜♠♥♠➈❲♦❘❥❣❜❦❥ú❳ê ❦❚❘❥❤❥❲♦❨➽❩❭❣❳ò❇❲✫❛❜❤❚❤❥❙❱❪❬❙♥❣❳❘❉❛q♠éê⑨î➅ï▲❣❜❪❬ê❥❲♦❩❭❙♥❩õ
✐ô❪✙❙♥❩➽❛q❩❬❩❬❦❚ò❇❲✟❤ ❪❫ê❥❛q❪✯❪❫ê❥❲
❦❥❘❚ï▲❲✟❨❭❪❫❦❚❨❬ö➳❲♦❤➩❩❞î➅❩❭❪❫❲✟ò ê❥❛❜❩■❛❜❘➩❙❱❩❬❣❳♠♥❛q❪❫❲✟❤❃ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻✯❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘
õ
þ➯ê❥❲❐ï➳❲♦❨❞❪❫❦❥❨❭ö❉❛q❪❫❙❱❣❳❘✕❙♥❩
❩❭❦❥ï❥ï➳❣❳❩❬❲✟❤❑❪❫❣❷ö➳❲✚✙✜✛✣✢✥✤✧✦✜✛✣★✩★✫✪✭✬✮★✰✯✲✱✥✳✴✦✴✵✶✛✮✷✗✵✶✙✩❙♥❘✣❪❬ê❥❲➯❺➋❛❜❨✸✖➷❛❜❩✵❩❬❲✟❘❥❩❬❲✹✂❺ ✣☎ ♣❳❙
õ
❲
õ
❙➭❪P❙♥❩✵ï➳❲♦❨❬❙❱❣ø❤❚❙♥❻
ð
❙➭❪❫ê✫❛✯ï▲❲✟❨❬❙♥❣➅❤
ð
ê❥❙♥❻❹ê✫❻✌❛q❘✫ö➳❲❵❻❹ê❥❣❳❩❭❲♦❘☛❛qï❥ï❥❨❬❣❜ï❥❨❬❙♥❛q❪❫❲✟♠❱î
õ
✎
❲❵❩❬ê❉❛❜♠❱♠✮ï❥❨❭❣✻❯❜❲✣❛q❘✫❲✰÷ø❙♥❩❭❪❬❲♦❘❥❻✟❲■❪❬ê❥❲♦❣❜❨❬❲♦ò✜í➐❣❳❨✲❪❫ê❥❙❱❩ó❲♦ì❿❦❉❛q❪❬❙♥❣❳❘
õ
r✩❣❳❦❥❤✺✂r✻☎●❛❜♠♥❨❭❲✌❛❜❤øî❇ï❚❨❬❣➠❯❳❲✟❩óí➐❣❜❨é❪❫ê❥❲✇❻✌❛❜❩❭❲
f
(
x
)
≡
c
♣❚❪❬ê❥❲❷❲❸÷ø❙❱❩❭❪❫❲✟❘❥❻♦❲❵❣❜í✵❛❑ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻ ❩❭❣❳♠♥❦❚❪❬❙♥❣❳❘✡❣qí✝❪❬ê❥❲❵❲♦ì❿❦❉❛q❪❬❙♥❣❳❘x
′′
+
cx
′
+
g
(
x
) =
ǫh
(
t
)
,
✡✄
✌
ð
ê❥❲✟❨❬❲✕❪❫ê❥❲❃ï➳❲♦❨❭❪❬❦❥❨❬ö❥❛q❪❫❙❱❣❳❘ ❤❥❣ø❲✟❩✛❘❚❣❜❪✫❤❥❲♦ï➳❲♦❘❥❤ ❣❳❘ ❪❬ê❥❲❃❩❭❪❹❛➷❪❫❲
õ
☎❲✕❦❥❩❭❲♦❩✛í➐❣❜❨æ❪❬ê❉❛q❪
❛➩❯q❛❜❨❭❙↔❛❜❘⑨❪✡❣❜í❵❪❬ê❥❲ù❙♥ò✙ï❥♠♥❙❱❻♦❙➭❪✙í➐❦❚❘❥❻✟❪❬❙♥❣❳❘✂❪❬ê❥❲♦❣❳❨❭❲♦ò
õ
②✫❣❳❨❬❲ù❲✰÷❥❛q❻✟❪❫♠➭î❳♣tê❥❲✕❻♦❣❳❘❚❩❬❙♥❤❚❲♦❨❬❩æ❛
í➐❦❥❘❚❻✟❪❫❙❱❣❳❘
g
(
x
) =
xk
(
x
)
ð ê❥❲♦❨❭❲k
❙❱❩✣❻♦❣❜❘❿❪❬❙♥❘❿❦❥❣❳❦❚❩❬♠❱î✕❤❚❙❱ë▲❲♦❨❬❲✟❘⑨❪❫❙↔❛qö❥♠♥❲❇❛❜❘❥❤k
(
x
)
>
0
,
x
6= 0
õ②✛❣❳❨❭❲♦❣➠❯❳❲✟❨♦♣
x
d
dx
k
(
x
)
>
0
,
x
6= 0
❣❳❨
x
d
dx
k
(
x
)
<
0
,
x
6= 0
❛❜♠
ð
❛✌î➅❩✇ê❚❣❳♠♥❤❥❩
ð
❙❱❪❬ê✛❪❬ê❥❲✣ï➳❣❳❩❬❩❭❙♥ö❥♠❱❲❵❲❸÷ø❻♦❲✟ï❚❪❫❙❱❣❳❘☛❣❜í✗❙❱❩❬❣❳♠♥❛q❪❫❲✟❤✛ï➳❣❳❙❱❘❿❪❬❩
õ
❡☎❣q❪❫❙♥❻✟❲■❪❬ê❉❛q❪t❪❫ê❥❲
❛❜ö➳❣➠❯❳❲❷❻✟❣❳❘❥❤❥❙➭❪❫❙❱❣❳❘❥❩❅❙♥ò✙ï❥♠❱î✯❣❜❘❀❣❳❘❥❲☎ê❉❛❜❘❥❤❀❪❬ê❥❲✇ò✙❣❳❘❥❣❜❪❬❣❳❘❥❙❱❻♦❙❱❪❢î❑❣❜í✩❪❫ê❥❲☎ï▲❲✟❨❬❙❱❣ø❤✙í➐❦❥❘❥❻✟❪❬❙♥❣❳❘
T
í➐❣❳❨❷❪❫ê❚❲✯❩❭î➅❩❭❪❬❲♦òx
′′
+
g
(
x
) = 0
õ✘✎ ê❥❲♦❘
g
❙♥❩✖❙♥❘✕❛q❤❥❤❥❙❱❪❬❙♥❣❳❘✒❤❥❙❱ë▲❲♦❨❭❲♦❘⑨❪❫❙♥❛❜ö❥♠♥❲■❪❫ê❥❲✟❩❬❲ ❻✟❣❳❘❥❤❥❙➭❪❫❙♥❣❜❘❥❩✲❙♥ò✙ï❥♠➭î❐❣❳❘✡❪❬ê❥❲❵❣❜❪❫ê❥❲✟❨óê❉❛❜❘❥❤g
′′
(0) = 0
õ r✩❲✟❪u
(
t
)
ö▲❲❵❛➽❘❥❣❳❘ø❝ô❻♦❣❳❘❚❩❭❪❹❛q❘❿❪ω
❝ôï▲❲✟❨❬❙♥❣➅❤❥❙❱❻☎❩❭❣❳♠♥❦❚❪❬❙♥❣❳❘✡❣qí✝❪❬ê❥❲❵❲♦ì❿❦❉❛q❪❬❙♥❣❳❘x
′′
+
g
(
x
) = 0
F
(
s
) =
Z
∞
0
u
′
(
t
+
s
)
f
(
t
)
dt.
s☎♠♥❩❬❣❚♣ ✂r ☎✬❣❳ö❥❩❭❲♦❨❭❯❜❲♦❩❵❪❫ê❉❛➷❪✖❙❱íPí➐❣❜❨❷❩❬❣❳ò✙❲
s
0
,
F
(
s
0
) = 0
ð
ê❥❙♥♠❱❲
F
′
(
s
0
)
6= 0
❪❫ê❥❲✟❘
í➐❣❳❨✇❩❬❦ ✁ ❻♦❙❱❲♦❘⑨❪❫♠➭îù❩❬ò❇❛❜♠❱♠
ǫ >
0
❪❬ê❥❲♦❨❭❲✯❲❸÷ø❙♥❩❞❪❫❩■❛❜❘ω
❝ôï▲❲✟❨❬❙❱❣ø❤❥❙❱❻ ❩❭❣❳♠❱❦❚❪❫❙❱❣❳❘v
(
t, ǫ
)
❣❜í ❪❬ê❥❲❵ï▲❲✟❨❭❪❫❦❚❨❬ö➳❲♦❤✫❲♦ì❿❦❉❛q❪❬❙♥❣❳❘x
′′
+
g
(
x
) =
ǫf
(
t
) =
ǫf
(
t
+
ω
)
✂ ✄✆☎❐è✞✝✩á✠✟óàæç✡✟☞☛ àæä à✡ã à✍✌✎✟✡☎❐è✏✝✮á✠✟óàæç✡✟✦ã✒✑✔✓✔✟✲â✝è✟ã✙ä✡è♦ç✕✝✩ã✒✖♦å✍✌
á●è✟ã à✗✝ ã✒✑
E
ǫ
✘✚✙✜✛ ✢ ✣✥✤✦✣★✧✪✩✬✫✮✭✰✯✱✧✲✣✥✳✴✧✶✵✎✧✲✭✸✷✥✹✺✯
s➯❻✟❻♦❣❳❨❭❤❥❙♥❘❥ú■❪❫❣✣❴
õ
✇❛❜❨❞❪❫ò❇❛❜❘☛✡ ✂ ☎ ♣➅ï
õ
⑥❳⑦✭✌❸♣❿❲✟ì➅❦❥❛q❪❫❙❱❣❳❘ ✡
1
ǫ
✌✗❙♥❘✙ú❳❲✟❘❥❲♦❨❬❛❜♠▲❤❚❣ø❲✟❩➈❘❥❣q❪éê❉❛✌❯❜❲
❛✯❘❥❣❳❘✫❻♦❣❜❘❥❩❭❪❫❛❜❘⑨❪☎ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻✇❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘✮♣❚❲❸❯❳❲✟❘✒❙❱í
xg
(
t, x, x
′
)
>
0
õ þ➯ê❥❲✫í➐❣❳♠♥♠❱❣ð
❙♥❘❥úù❲✰÷❥❛qò❇ï❥♠❱❲✛ú❜❙❱❯❳❲✟❘ ö⑨î ②✛❣❳❩❭❲♦❨❐ï❚❨❬❣➠❯❳❲✟❩❐❪❫ê❥❲✛❘❥❣❳❘ ❲❸÷ø❙♥❩❞❪❫❲♦❘❚❻♦❲☛❣❜í✖❛ ❘❥❣❳❘
❻✟❣❳❘❥❩❭❪❫❛❜❘⑨❪tï▲❲✟❨❬❙♥❣➅❤❥❙❱❻❷❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘✡❣❜í
x
′′
+
φ
(
t, x, x
′
) = 0
.
r✩❲✟❪
φ
(
t, x, y
) =
x
+
x
3
+
ǫf
(
t, x, y
)
,
ǫ >
0
❩❬❛q❪❫❙❱❩❭íî➅❙♥❘❥ú➈❪❫ê❥❲✗í➐❣❳♠❱♠♥❣
ð
❙♥❘❥ú➈❻♦❣❳❘❚❤❥❙❱❪❬❙♥❣❳❘❥❩➳í➐❣❳❨
φ
∈
C
1
(
R
3
)
,
f
(
t
+1
, x, y
) =
f
(
t, x, y
)
,
ð
❙➭❪❫ê
f
(0
,
0
,
0) = 0
,
f
(
t, x, y
) = 0
if
xy
= 0
φ
x
→ ∞
when
x
→ ∞
❦❥❘❚❙❱í➐❣❳❨❭ò❇♠➭î❇❙❱❘
(
t, y
)
∈
R
2
,
δf
δy
>
0
if
xy >
0
,
and
δf
δy
= 0
otherwise
.
x, y
❯❜❲♦❨❭❙❱í î➅❙♥❘❥ú|
x
|
< ǫ,
|
y
|
< ǫ.
✐❢❘✣í❖❛q❻✟❪✌♣
ð
❲éê❉❛✌❯❳❲
xf
(
t, x, y
)
❛❜❘❥❤yf
(
t, x, y
))
>
0
❙➭íxy >
0
,
|
x
|
< ǫ, y
❛❜❨❬ö❥❙➭❪❫❨❬❛❜❨❭î ❛❜❘❚❤φ
= 0
❣q❪❫ê❥❲✟❨ð
❙♥❩❭❲
õ
þ➯ê❥❲tí➐❦❥❘❥❻✟❪❬❙♥❣❳❘
V
= 2
x
2
+
x
4
+ 2
x
′
2
❩❫❛q❪❬❙♥❩ ✓❥❲♦❩V
′
=
−4
ǫx
′
f
(
t, x, x
′
)
♣➅❩❬❣ ❪❬ê❉❛q❪V
′
<
0
❙❱íxx
′
>
0
,
|
x
|
< ǫ
❛❜❘❥❤V
′
= 0
❣❜❪❬ê❥❲♦❨ð
❙♥❩❬❲
õ
þ➯ê❿❦❥❩
x
❻✌❛❜❘❚❘❥❣❜❪tö▲❲❵ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻❷❦❚❘❥♠♥❲✟❩❬❩V
′
= 0
õþ➯ê❥❙❱❩ ❲✰÷❥❛qò❇ï❥♠❱❲➽❙♥❩❵❩❬❙❱ú❳❘❥❙✓❥❻✌❛❜❘⑨❪❵ö➳❲♦❻✌❛q❦❥❩❬❲❇❙❱❪❵❩❬ê❚❣
ð
❩✚✖❿❙♥❘❥❤ ❣❜í✲❤❥❙✁ ❻♦❦❥♠➭❪❫❙♥❲✟❩✖❲♦❘❥❻✟❣❳❦❥❘ø❝
❪❬❲♦❨❬❲✟❤✡❪❫❣➽❲♦❩❞❪❹❛❜ö❥♠❱❙♥❩❭ê✡❲❸÷ø❙♥❩❞❪❫❲♦❘❚❻♦❲■❨❬❲♦❩❭❦❥♠❱❪❬❩➯❣❜í✵ï▲❲✟❨❬❙❱❣ø❤❥❙❱❻❷❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘❥❩❅í➐❣❳❨óû✗ì❿❦❉❛q❪❫❙❱❣❳❘
Eǫ
õ þ✝❣❪❬ê❉❛q❪❷❲✟❘❥❤✮♣➳❪❫ê❥❲❑❲✰÷ø❙♥❩❭❪❬❲♦❘❥❻✟❲❑ï❥❨❭❣❳ö❥♠♥❲✟ò
ð
❙♥♠♥♠●ö➳❲❑ò✙❣❳❨❬❲■❻♦❣❳❘⑨❯❜❲♦❘❥❙❱❲♦❘⑨❪✖❪❫❣❀❩❞❪❫❦❥❤øî
ð
ê❥❲✟❘✕❛❜❤ø❝
❤❥❙➭❪❫❙❱❣❳❘❉❛❜♠✵ê⑨îøï➳❣❜❪❬ê❥❲♦❩❭❲♦❩■❣❳❘ù❪❫ê❥❲➽ï➳❲♦❨❬❙❱❣ø❤✕❛q❨❬❲✯❨❬❲✟ì❿❦❥❙♥❨❭❲♦❤
õ
❺❉❣❳❨❷❲✰÷❚❛❜ò❇ï❚♠♥❲❜♣✮❪❬ê❥❲✯ï➳❲♦❨❬❙❱❣ø❤ù❣qí
❪❬ê❥❲❑ï➳❲♦❨❭❪❬❦❥❨❬ö➳❲♦❤✕r✩❙❱❲♦❘❉❛❜❨❭❤☛❲♦ì❿❦❉❛➷❪❫❙♥❣❜❘
x
′′
+
f
(
x
)
x
′
+
g
(
x
) =
ǫh
(
t, x, x
′
, ǫ
)
.
ê❉❛❜❩✇❪❬❣ ö➳❲✁❻✟❣❳❘⑨❪❫❨❬❣❜♠♥♠♥❲✟❤✂❚❙❱❘æ❣❳❨❬❤❥❲✟❨➯❪❫❣➽❩❭❪❫❛q❪❫❲❵❲❸÷ø❙❱❩❭❪❫❲✟❘❥❻♦❲■❣❜í✝ï▲❲✟❨❬❙❱❣ø❤❥❙❱❻❷❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘õ
✘✚✙ ✘ ✄✆☎✦✭✸✧✞✝✠✟ ✧ ✵✎✧
g
✫✮✭ ✫✮✣☛✡✥✧✂☞ ✧✲✣☛✡ ✧✲✣ ✯ ✤✍✌x
′
✉é❣❳❘❚❩❬❙♥❤❚❲♦❨t❛❜❘æ❲♦ì❿❦❉❛q❪❬❙♥❣❳❘æ❣❜í✵❪❫ê❚❲❵❪❢î➅ï➳❲
x
′′
+
φ
(
t, x, ǫ
) = 0
,
✡ ✌ð
ê❥❲✟❨❬❲
ǫ >
0
❙❱❩é❛ ❩❬ò❇❛❜♠♥♠➋ï❉❛❜❨❬❛❜ò✙❲✟❪❫❲✟❨♦♣φ
❙❱❩é❛✣❻✟❣❳❘⑨❪❫❙❱❘➅❦❚❣❳❦❥❩éí➐❦❥❘❚❻✟❪❫❙❱❣❳❘✮♣T
−
ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻ ❙❱❘t
❩❬❦❚❻❹ê✛❪❫ê❥❛q❪φ
(
t, x,
0) = ˜
g
(
x
)
õ ②✛❣❳❨❭❲❷ï❥❨❭❲♦❻♦❙❱❩❬❲✟♠❱î❳♣❚❦❥❘❥❤❥❲✟❨é❪❫ê❥❲❷í➐❣❜♠♥♠♥❣ð
❙❱❘❥ú ê⑨î➅ï➳❣❜❪❫ê❥❲✟❩❬❲✟❩óí➐❣❜❨➈❪❫ê❚❲❷í➐❦❥❘❥❻✟❪❬❙♥❣❳❘
g
❤❥❲✔✓❉❘❚❲♦❤ ❣❳❘R
×
(
α, β
)×]0
, ǫ
0
]
.
✎
(1)
φ
is T
−
periodic on t
(2)
φ
(
t, x,
0) = ˜
g
(
x
)
(3)
if
x
6= 0
,
we have
g
˜
(
x
)
x >
0
.
✡ ⑥✭✌
þ➯ê❉❛q❪tò✙❲✌❛❜❘❚❩✲í➐❣❳❨
ǫ
= 0
❪❫ê❥❲■❛❜❦❚❪❬❣❳❘❥❣❳ò✙❣❳❦❥❩➯❩❞î➅❩❭❪❫❲✟òx
′
=
y
y
′
=
−˜
g
(
x
)
✡❖⑧ ✌
ê❉❛❜❩➯❪❬ê❥❲❵❣❳❨❬❙❱ú❳❙♥❘
(0
,
0)
❛❜❩t❛✯❻♦❲♦❘⑨❪❬❲♦❨õ
þ➯ê❥❙❱❩❷ò✙❲✌❛❜❘❥❩☎❪❫ê❥❲✑✏❉❣
ð
❙♥❘❥❤❚❦❥❻♦❲✟❤❃ö⑨î✛❪❫ê❥❲❑❯❜❲♦❻✟❪❬❣❳❨ ✓❉❲✟♠♥❤ù❣❜í✑❪❫ê❚❲ ✇❛❜ò✙❙♥♠➭❪❫❣❳❘❚❙↔❛❜❘✫❩❭î➅❩❭❪❬❲♦ò
✡➐⑧ ✌➈ê❉❛❜❩ó❛✯❩❞❪❹❛q❪❬❙♥❣❳❘❉❛q❨❭î❇ï▲❣❳❙❱❘⑨❪➯❛q❪➯❪❬ê❥❲✖❣❳❨❬❙❱ú❳❙♥❘❐❛❜❘❥❤æ❙♥❩é❩❬❦❥❨❭❨❬❣❳❦❥❘❚❤❥❲♦❤✫ö⑨îæ❛✣í❖❛❜ò✙❙♥♠❱î✙❣❜í✵ï➳❲❸❝
❨❭❙♥❣➅❤❥❙♥❻ó❣❳❨❭ö❥❙❱❪❬❩
õ
û✑❛❜❻❹ê✡❣❳❨❭ö❥❙❱❪
γ
❣❜í✩❪❬ê❥❙♥❩✑í❖❛❜ò✙❙♥♠➭î✯♠♥❙❱❲♦❩❅❣❳❘❀❛❜❘❐❲♦❘❥❲✟❨❬ú❜î❇♠♥❲❸❯❳❲♦♠ ♣ø❩❬❛✌îc
♣❀❛❜❘❚❤γ
≡
γ
(
c
)
.
þ➯ê❚❲✣ï➳❲♦❨❬❙❱❣ø❤✡í➐❦❚❘❥❻✟❪❬❙♥❣❳❘T
(
c
)
❤❥❲♦ï➳❲♦❘❚❤❥❙♥❘❥ú❇❣❳❘c
❙❱❩ó❪❬ê❥❲■ò❇❙❱❘❥❙♥ò❇❛❜♠➳ï▲❲✟❨❬❙♥❣➅❤ ❣❜í✎❪❫ê❥❙❱❩é❣❳❨❭ö❥❙❱❪õ ✎ ❲☎❩❫❛✌î
T
❙❱❩➈ò✙❣❳❘❥❣q❪❫❣❳❘❥❲☎❙❱í✮❪❬ê❥❲☎í➐❦❚❘❥❻✟❪❬❙♥❣❳❘T
(
c
)
❙♥❩➈ò❇❣❜❘❥❣❜❪❫❣❜❘❥❲õ
þ➯ê❚❲
❤❥❲✟ï▲❲✟❘❥❤❥❲✟❘❥❻♦❲❷❣qí✮❪❫ê❥❲tï➳❲♦❨❬❙❱❣ø❤✙❣❳❘✙❪❫ê❥❲☎❲♦❘❥❲✟❨❬ú❜î❇❛❜❘❥❤✙❪❫ê❥❲tò✙❣❳❘❥❣❜❪❬❣❳❘❥❙♥❻✟❙❱❪❢î✣❻♦❣❳❘❥❤❚❙❱❪❫❙❱❣❳❘❥❩✑ê❉❛❜❩
ö➳❲♦❲✟❘✡❩❭❪❫❦❚❤❥❙♥❲✟❤❀ö⑨î❀❛✣❘❿❦❥ò✣ö▲❲✟❨✲❣❜í✵❛❜❦❚❪❬ê❥❣❳❨❭❩
õ
❺❉❣❳❨✑❪❫ê❥❲✟❩❬❲✖ì❿❦❥❲♦❩❞❪❫❙❱❣❳❘❥❩♦♣
ð
❲✖❨❭❲✟í➐❲✟❨➈❪❫❣✕✂✉❅❝➦✉ ☎
õ
✎ õ õ
❛❜❘❚❤✛s
õ
❺❉❣❳❘❚❤❉❛ ✂✠❅❝❢❺ ☎●ï❥❨❬❣➠❯❜❲♦❤✒❪❬ê❉❛q❪ó❪❫ê❥❲✟❨❬❲ ❛❜❨❬❲■ï➳❲♦❨❬❙❱❣ø❤❚❙♥❻✖❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘❥❩➯❣qí✆✡ ✗✌➯ï❚❨❬❣➠❯➅❙♥❤❥❲✟❤
❪❬ê❉❛q❪➯❪❫ê❚❲ ï➳❲♦❨❭❙♥❣➅❤❀í➐❦❥❘❥❻✟❪❬❙♥❣❳❘
T
=
T
(
c
)
❣qí✝❪❬ê❥❲✣❛q❦❚❪❫❣❳❘❚❣❳ò❇❣❜❦❥❩ó❛❜❩❬❩❭❣ø❻✟❙↔❛q❪❬❲♦❤æ❩❭î➅❩❞❪❫❲♦ò ❙❱❩ ò✙❣❳❘❥❣❜❪❬❣❳❘❥❲t❛❜❘❥❤✙❪❫ê❉❛➷❪ǫ
❙♥❩✑❩❬ò❇❛❜♠♥♠❉❲✟❘❥❣❳❦❥ú❜êõ
φ
❙♥❩❅❛❜❩❭❩❬❦❥ò✙❲♦❤❐❪❬❣✣ö➳❲ ✡➐❣❳❘❥♠➭î ✌P❻✟❣❳❘⑨❪❫❙♥❘❿❦❥❣❜❦❥❩
õ
②✛❣❜❨❬❲✙ï❥❨❬❲✟❻♦❙♥❩❭❲♦♠➭î❳♣●❪❫ê❥❲❸î ❩❭ê❥❣
ð
❪❫ê❥❛q❪✣❙➭í➈❪❬ê❥❲➽í➐❦❥❘❥❻✟❪❬❙♥❣❳❘
φ
(
t, x, ǫ
)
❙❱❩■❻✟❣❳❘⑨❪❫❙❱❘➅❦❚❣❳❦❥❩♦♣✵❪❫ê❥❲✟❘ ❪❬ê❥❲✇ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻ó❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘❥❩❅❣❜í✩❩❭❦❥❻❹êæ❲♦ì❿❦❉❛q❪❬❙♥❣❳❘❚❩éò❇❛➠î➽ö➳❲✇♠❱❣ø❻♦❛q❪❫❲✟❤❐❘❥❲♦❛❜❨é❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘❥❩➈❣qí✮❪❫ê❥❲❛❜❦ø❪❫❣❳❘❥❣❜ò❇❣❳❦❚❩➯❲♦ì❿❦❉❛q❪❬❙♥❣❳❘✮♣❉ï❚❨❬❣➠❯➅❙♥❤❥❲✟❤✫❪❬ê❉❛q❪tï▲❲✟❨❬❙❱❣ø❤❥❙❱❻❷❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘❥❩➯❣qí ✡➐⑧ ✌✲❲❸÷ø❙♥❩❞❪✇❛q❘❥❤✫❪❬ê❉❛q❪
❪❬ê❥❲❵ï▲❲✟❨❬❙♥❣➅❤❀í➐❦❥❘❥❻❸❪❫❙❱❣❳❘æ❙♥❩➯❩❭❪❬❨❬❙❱❻✟❪❫♠➭î❐ò✙❣❳❘❥❣q❪❫❣❳❘❥❲
õ
②✛❣❳❨❭❲♦❣➠❯❳❲✟❨♦♣❉❪❫ê❥❲✟❨❬❲❵❙♥❩ó❛✯❩❭❣❳♠♥❦❚❪❬❙♥❣❳❘✡ò❇❛✗✖❳❝
❙❱❘❥ú✯❲❸÷❚❛❜❻❸❪❫♠❱î
N
❨❬❣❜❪❫❛q❪❫❙❱❣❳❘❥❩ó❛❜❨❭❣❳❦❥❘❥❤✡❪❫ê❚❲■❣❜❨❬❙♥ú❜❙♥❘❀❙❱❘æ❪❬ê❥❲✖❪❫❙❱ò❇❲kT.
þ➯ê❥❲✟❙♥❨■❨❬❲✟❩❬❦❥♠➭❪❫❩✣❙❱ò❇ï❚❨❬❣➠❯❳❲➽❪❫ê❥❣❜❩❬❲❐❣qíór✩❣❳❦❥❤ ✂r✻☎♣
ð
ê❥❣ù❛❜❩❭❩❬❦❥ò✙❲♦❤ ❪❫ê❉❛q❪■❪❫ê❚❲❇í➐❦❥❘❥❻❸❪❫❙❱❣❳❘
φ
ê❉❛q❤æ❪❬❣❇ö➳❲✖❻♦❣❜❘❿❪❬❙♥❘❿❦❥❣❳❦❚❩❬♠❱î❀❤❥❙❱ë▲❲♦❨❭❲♦❘⑨❪❫❙♥❛❜ö❥♠♥❲
õ
②✛❣❜❨❬❲♦❣➠❯❜❲♦❨♦♣t❙♥❘ ♠♥❙❱ú❳ê⑨❪❇❣qí✖❪❫ê❥❲☛ï❚❨❬❲♦❻✟❲♦❤❥❙❱❘❥úñ❲✰÷❥❛qò❇ï❥♠❱❲ ✂ ☎ ♣é❙➭❪❐❩❭❲♦❲✟ò❇❩❐❪❬ê❉❛q❪❇❪❫ê❥❲✛ò❇❲❸❪❫êø❝
❣➅❤❥❩■❤❥❲♦❩❭❻♦❨❬❙❱ö▲❲✟❤✺❛❜ö➳❣➠❯❳❲❐❤❚❣✒❘❥❣❜❪ ú❳❲✟❘❥❲♦❨❬❛❜♠♥❙✑♦❲➽❙❱íé❣❳❘❥❲❐❩❭❦❥ï❥ï➳❣❳❩❬❲✟❩
φ
❤❥❲♦ï➳❲♦❘❥❤❚❲♦❘⑨❪✯❣❳❘x
′
✎
φ
≡
φ
(
t, x, x
′
, ǫ
)
õ①ø❣❚♣✎❛❜❘❚❣❜❪❫ê❥❲✟❨☎❻✟❣❳❘❥❤❥❙➭❪❫❙❱❣❳❘✛❣❜❘✛❪❫ê❚❲✣ï➳❲♦❨❬❙❱❣ø❤✛❛❜ï❥ï➳❲✌❛q❨❬❩☎❪❬❣❐ö➳❲✣❘❥❲✟❻♦❲✟❩❬❩❫❛q❨❭î✫❪❫❣❐❣❜ö❚❪❹❛❜❙❱❘✒❲✰÷❚❙❱❩❞❝
❪❬❲♦❘❥❻✟❲■❣qíPï➳❲♦❨❬❙❱❣ø❤❚❙♥❻✇❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘❥❩➯❣❜í✵❪❫ê❥❲❵ï➳❲♦❨❭❪❬❦❥❨❬ö➳❲♦❤✫❲♦ì❿❦❉❛➷❪❫❙♥❣❜❘
õ
❡t❲✟❯❳❲✟❨❭❪❬ê❥❲♦♠❱❲♦❩❬❩✟♣✝❣❜❘❥❲❑❻✌❛q❘❃❩❭ê❥❣
ð
❛❜❘✕❛q❘❉❛❜♠♥❣❜ú❳❣❳❦❥❩☎❨❬❲♦❩❭❦❥♠❱❪✇❪❫❣❀❪❫ê❚❲✯ï❥❨❬❲✟❻♦❲♦❤❚❙♥❘❥úæ❣❳❘❥❲✯❦❚❘❥❤❥❲♦❨
ò✙❣❳❨❬❲❇❨❬❲✟❩❭❪❫❨❭❙♥❻❸❪❫❙❱❯❜❲❐ê⑨î➅ï▲❣q❪❫ê❥❲✟❩❬❲♦❩
õ
✠❅îñ❛❜❘ñ❛❜ï❥ï❥❨❭❣❳ï❥❨❬❙♥❛q❪❫❲✟❤ñ❻❹ê❥❣❳❙❱❻♦❲❀❣qí➯❪❫ê❥❲❀ï➳❲♦❨❭❙♥❣➅❤➩❣❜ít❛
ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻❷❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘✡❣❜í✝❪❬ê❥❲❵ï▲❲✟❨❭❪❬❦❥❨❬ö➳❲♦❤✛r✩❙♥❲♦❘❥❛❜❨❬❤æ❲♦ì❿❦❉❛➷❪❫❙♥❣❜❘
õ
✁ çéã à✯á✩â✬ã✒✖✞✖✞☛✄✂ ✖✆☎ ✓✔✟✲â✝è❸ã❇ä✡è✟ç ✓✔✟✲â✵á✵å✡â✝✂ ☛❵á●è❸ã à
✞ ❘❥❲❅❨❬❲❸í➐❲♦❨❬❩●❪❫❣☎❛☎ò✙❲✟❪❬ê❥❣➅❤ ❤❥❦❚❲❅❪❬❣✇❺➋❛❜❨ ✖q❛q❩✵❙❱❘❥❩❬ï❚❙♥❨❬❲✟❤ ö⑨î❵❪❫ê❚❲➈❣❳❘❚❲❅❣qí➋❴✝❣❳❙❱❘❥❻✌❛q❨✠✟
õ
þ➯ê❥❲➈❤❥❲❸❝
❪❬❲♦❨❬ò✙❙♥❘❥❛q❪❫❙❱❣❳❘❑❣❜í➳❻♦❣❜❘❿❪❬❨❬❣❳♠❱♠↔❛❜ö❚♠❱î■ï▲❲✟❨❬❙❱❣ø❤❥❙❱❻✲ï➳❲♦❨❭❪❬❦❥❨❬ö➳❲♦❤✙❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘
õ
þ➯ê❥❙❱❩❅ò✙❲✟❪❬ê❥❣➅❤✯ï❥❨❬❣➠❯❳❲✟❤
❪❬❣ ö➳❲✛❙➭❪❫❩❭❲♦♠❱í☎❯❳❲✟❨❭î ❲✟ë▲❲♦❻❸❪❫❙❱❯❜❲✒ï❉❛q❨❭❪❫❙❱❻♦❦❥♠♥❛❜❨❬♠➭î í➐❣❜❨➽❪❬ê❥❲✒ï➳❲♦❨❞❪❫❦❥❨❭ö❉❛q❪❬❙♥❣❳❘❥❩➽❣❜í☎❯q❛❜❨❬❙❱❣❳❦❥❩✙❛❜❦ø❝
❪❬❣❳❘❥❣❳ò✙❣❳❦❥❩☎❩❭î➅❩❭❪❬❲♦ò✙❩
õ✚✎ ❲ ✖❿❘❥❣
ð
í➐❣❜❨❷❲❸÷❚❛❜ò✙ï❥♠♥❲❑❛❀ú❳❣ø❣➅❤ù❛❜ï❚ï❥♠♥❙❱❻✌❛q❪❬❙♥❣❳❘æí➐❣❳❨❷ï➳❲♦❨❞❪❫❦❥❨❭ö▲❲✟❤
❶❅❛❜❘ù❤❚❲♦❨❵❴✝❣❜♠P❲♦ì❿❦❉❛q❪❬❙♥❣❳❘❚❩❷❪❢îøï➳❲ ✂❺ ✣☎
õ
þ➯ê❥❲✯ï➳❲♦❨❞❪❫❦❥❨❭ö❉❛q❪❬❙♥❣❳❘ù❙❱❩❷❩❬❦❥ï❚ï▲❣❳❩❭❲♦❤✕❪❫❣✡ö➳❲☛✡❻✟❣❳❘ø❝
❪❬❨❬❣❳♠❱♠↔❛❜ö❥♠➭î ï▲❲✟❨❬❙♥❣➅❤❥❙❱❻ ➭♣t❙
õ
❲
õ
♣☎❙❱❪✡❙❱❩✡ï▲❲✟❨❬❙❱❣ø❤❥❙❱❻
ð
❙❱❪❬ê ❛ ï➳❲♦❨❭❙♥❣➅❤
ð
ê❥❙♥❻❹ê ❻✌❛❜❘ ö➳❲✕❻❹ê❥❣❳❩❬❲✟❘
❛❜ï❚ï❥❨❬❣❳ï❚❨❬❙↔❛➷❪❫❲♦♠➭î
õ
◗☎❘❥❤❚❲♦❨❑❯❳❲✟❨❭î✺ò✙❙♥♠❱❤ñ❻♦❣❜❘❥❤❥❙❱❪❬❙♥❣❳❘❚❩❑❙❱❪✯❙❱❩❑ï❥❨❭❣✻❯❜❲♦❤ ❪❫ê❉❛➷❪✯❪❫❣ù❲✌❛q❻❹ê ❩❭ò❐❛❜♠❱♠
❲✟❘❥❣❳❦❥ú❳ê✛❛❜ò✙ï❥♠♥❙➭❪❫❦❥❤❚❲✇❣❜í✵❪❫ê❥❲❵ï➳❲♦❨❞❪❫❦❥❨❭ö❉❛q❪❫❙❱❣❳❘✡❪❫ê❥❲✟❨❬❲❵ö➳❲♦♠♥❣❜❘❥ú❳❩ó❛✯❣❳❘❥❲❵ï❉❛❜❨❬❛❜ò✙❲✟❪❫❲✟❨✲í❖❛❜ò✙❙♥♠❱î
❣❜í✇ï▲❲✟❨❬❙❱❣ø❤❥❩➽❩❭❦❥❻❹ê✂❪❫ê❉❛➷❪✙❪❬ê❥❲✒ï➳❲♦❨❞❪❫❦❥❨❭ö▲❲✟❤ ❩❞îø❩❞❪❫❲✟ò ê❉❛q❩❐❛❃❦❥❘❥❙♥ì❿❦❥❲✫ï➳❲♦❨❬❙❱❣ø❤❚❙♥❻✡❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘
ð
❙➭❪❫ê✡❪❫ê❥❙❱❩➯ï▲❲✟❨❬❙❱❣ø❤
õ
✎
❲❀♠♥❣❜ú❳❙♥❻♦❛❜♠♥♠➭î☛ò❇❛➠î❃❲✰÷❚ï➳❲♦❻❸❪✯❪❫ê❉❛➷❪✣❪❫ê❚❲✡❺➋❛❜❨✸✖➷❛❜❩ ò❇❲❸❪❫ê❥❣➅❤ñ❻✌❛q❘✺ö▲❲✡❛qú⑨❛❜❙♥❘➩❛❜ï❥ï❥♠❱❙♥❲✟❤ í➐❣❳❨
ï➳❲♦❨❞❪❫❦❥❨❭ö▲❲✟❤✺r✩❙♥❲♦❘❥❛❜❨❬❤ñ❲✟ì❿❦❉❛q❪❫❙❱❣❳❘❥❩
õ
þ➯ê❥❙❱❩✣ê❉❛❜❩ ö▲❲✟❲♦❘✺❻✟❣❳❘❥❩❬❙❱❤❥❲♦❨❭❲♦❤ ❛❜❘❚❤ñï❥❨❬❣➠❯❜❲♦❤✺❙♥❘❃❪❫ê❥❲
❛❜❦ø❪❫❣❳❘❥❣❜ò❇❣❳❦❚❩➯❻✌❛❜❩❭❲■ö⑨îæ❺➋❛❜❨ ✖➷❛❜❩➯ê❥❙♥ò✙❩❬❲✟♠❱í ✂❺
✄
☎
õ
✞ ❦❥❨✝ï❥❨❬❣➅❣❜í
ð
❲✲ú❳❙❱❯❜❲éê❥❲✟❨❬❲➯ò❇❛❜❤❥❲➈❙❱❪✝ò❇❣❳❨❭❲é❩❭❙♥ò✙ï❥♠♥❲❅❛❜❘❥❤✯❻✟❣❳❘⑨❪❹❛❜❙❱❘❥❩✬❩❭❣❳ò✙❲✲❙♥ò✙ï❥❨❬❣➠❯❜❲♦ò✙❲♦❘⑨❪❫❩
✄
❣➅❤❥❙♥❻✇❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘❥❩✲í➐❣❳❨✲❪❬ê❉❛q❪ó❲♦ì❿❦❉❛➷❪❫❙♥❣❜❘
õ
❡t❣❜❪❫❙❱❻♦❲❀❪❫ê❥❛q❪✯❪❫ê❥❲æ❩❬❛❜ò❇❲æï❥❨❭❣❳ö❥♠❱❲♦ò ê❥❛❜❩➽ö▲❲✟❲♦❘ ❻♦❣❳❘❥❩❭❙♥❤❥❲✟❨❬❲✟❤ ö➳❲✟í➐❣❳❨❭❲✫❙❱❘ñ❪❫ê❥❲æï❉❛qï▲❲✟❨✙❣qí
❺➋❛❜❨ ✖q❛q❩ó❛❜❘❥❤æs✇ö➳❤❥❲♦♠✁❵❛❜❨❭❙♥ò ✂❺✵❝ôs ☎✩ö❚❦❚❪t❣❳❦❥❨➯❨❬❲✟❩❬❦❥♠➭❪❫❩t❛❜❨❭❲■ò✙❣❳❨❭❲❷ú❳❲♦❘❚❲♦❨❫❛q♠
õ
✂✬✙✜✛ ✄✆☎✦✭✱✫✮✳ ✟✆☎ ☞ ✤ ✯ ✟✥✧✲✭✸✧✲✭
r✩❲✟❪t❦❥❩ó❻✟❣❳❘❥❩❬❙❱❤❥❲♦❨✲❪❬ê❥❲ ✡❖❦❥❘❚ï▲❲✟❨❭❪❫❦❚❨❬ö➳❲♦❤✍✌ór✩❙♥❲♦❘❥❛❜❨❬❤æ❲♦ì❿❦❉❛➷❪❫❙♥❣❜❘
(
L
)
u
′′
+
f
(
u
)
u
′
+
g
(
u
) = 0
.
✐❢❘æ❣❳❨❭❤❥❲♦❨➯❪❫❣➽ê❉❛✌❯❜❲ ❛✯❦❥❘❚❙♥ì❿❦❥❲❵ï▲❲✟❨❬❙❱❣ø❤❥❙❱❻❷❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘
ð
❲■❩❬❦❥ï❥ï➳❣❳❩❭❲❵❪❬ê❥❲✖í➐❦❥❘❥❻❸❪❫❙♥❣❜❘❥❩
f
❛❜❘❚❤g
❛❜❨❬❲✖❣❜í✝❻♦♠♥❛❜❩❬❩C
2
õ
þ➯ê❥❲ ❙❱❘⑨❪❫❲♦ú❜❨❫❛❜♠❱❩
F
(
x
) =
Z
x
0
f
(
t
)
dt,
G
(
x
) =
Z
x
0
g
(
t
)
dt
❣❜í
f
❛❜❘❚❤g
❨❬❲✟❩❬ï➳❲♦❻❸❪❫❙❱❯❜❲♦♠➭î■❛❜❨❬❲✗❩❭❦❥❻❹ê ❪❬ê❉❛q❪limx
→∞
F
(
x
) =
∞
,
❛❜❘❥❤
limx
→∞
G
(
x
) =
∞
.
✐ô❪■❙❱❩❵❛❜❩❬❩❭❦❥ò❇❲✟❤❃❪❬ê❉❛q❪F
ê❉❛❜❩■❛✡❦❥❘❥❙♥ì❿❦❥❲✞✑✟❲♦❨❬❣õ
þ➯ê❚❲♦❘❃❙❱❪❵❙♥❩ ✖❿❘❥❣
ð
❘☞✂✉❅❝❢r✻☎✗❪❫ê❉❛q❪
✡❖r ✌éê❥❛❜❩t❛➽❩❞❪❹❛❜ö❚♠♥❲❵❘❥❣❳❘æ❻♦❣❜❘❥❩❭❪❫❛❜❘⑨❪✇ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻☎❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘
u
0
(
t
)
ð
❙❱❪❬ê✛ï➳❲♦❨❭❙♥❣➅❤
τ
0
õ û✗ì❿❦❉❛q❪❬❙♥❣❳❘ ✡ r ✌✬❙❱❩✗❦❥❩❬❦❉❛q♠♥♠❱î ❩❭❪❬❦❥❤❥❙❱❲♦❤✙ö❿î❑ò❇❲♦❛❜❘❥❩✗❣❜í✎❛❜❘➽❲♦ì❿❦❥❙➭❯❜❛q♠♥❲♦❘⑨❪✑ï❥♠♥❛❜❘❥❲➯❩❞îø❩❞❪❫❲✟òõ
þ➯ê❥❲
ò✙❣❳❩❭❪ó❦❚❩❬❲♦❤✫❣❳❘❥❲✟❩✇❛q❨❬❲
✎
u
′
=
v
v
′
=
−
g
(
u
)
−
f
(
u
)
v
✡ ⑤ ✌
❛❜❘❥❤✛❛❜♠❱❩❬❣
u
′
=
v
−
F
(
u
)
v
′
=
−
g
(
u
)
✡✞✝✭✌
✐❢❘æí❖❛❜❻✟❪♦♣❚❪❫ê❥❲❸î✫❛❜❨❬❲✖❲✟ì❿❦❥❙❱❯q❛❜♠❱❲♦❘⑨❪ó❪❫❣✯❪❫ê❥❲ ✻❝ô❤❥❙♥ò✙❲♦❘❚❩❬❙♥❣❜❘❉❛❜♠➳❩❭î➅❩❭❪❬❲♦ò
(
S
)
x
˙
=
h
(
x
)
❛qí❪❬❲♦❨➯❙❱❘❿❪❬❨❬❣➅❤❥❦❥❻✟❙♥❘❥ú❑❪❬ê❥❲❵❘❥❣❜❪❹❛➷❪❫❙♥❣❜❘❥❩
x
=
col
[
x
1
, x
2
]
x
1
=
−
u
˙
(
t
)
−
F
(
u
(
t
))
x
2
=
u
(
t
)
✡ ③ ✌
ð ê❥❲✟❨❬❲
x
=
col
[
x
1
, x
2
]
❛❜❘❥❤
h
(
x
) =
col
[
g
(
x
2
)
,
−
x
1
−
F
(
x
2
(
t
))
.
①ø❦❚ï❥ï▲❣❜❩❬❲
u
0
(0) =
a,
u
′
0
(0) = 0
>
0
❩❭❣✙❪❬ê❉❛q❪➯❪❫ê❚❲■ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻✇❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘✡❣❜í✝ï▲❲✟❨❬❙♥❣➅❤
τ
0
❣❜í✝❪❫ê❥❲❷❯q❛❜❨❭❙↔❛q❪❬❙♥❣❳❘❥❛❜♠▲❩❞î➅❩❭❪❫❲✟ò
˙
y
=
h
′
x
(
p
(
t
))
y
❙❱❩
˙
p
(
t
) =
col
[
g
(
u
0
(
t
))
,
u
˙
0
(
t
)]
,
ð
ê❥❲✟❨❬❲
p
(
t
) =
col
[−
u
˙
0
(
t
)
−
F
(
u
0
(
t
))
, u
0
(
t
)]
.
①ø❣❥♣❥❪❬ê❥❲❵❙♥❘❥❙➭❪❫❙♥❛❜♠➋❻♦❣❳❘❥❤❚❙❱❪❫❙❱❣❳❘❥❩ó❛❜❨❭❲
p
(0) =
col
[−
F
(
a
)
, a
]
,
p
˙
(0) =
col
[
g
(
a
)
,
0]
.
r✩❲✟❪t❦❥❩ó❻♦❣❜❘❥❩❬❙❱❤❥❲♦❨➯❪❫ê❚❲❵í➐❣❜♠♥♠♥❣
ð
❙❱❘❥ú❑ï➳❲♦❨❭❪❬❦❥❨❬ö➳❲♦❤✛r✩❙♥❲✟❘❉❛❜❨❬❤æ❲✟ì➅❦❥❛q❪❫❙❱❣❳❘æ❣❜í✝❪❬ê❥❲✖í➐❣❳❨❬ò
(
LR
)
u
¨
+
f
(
u
) ˙
u
+
g
(
u
) =
ǫγ
(
t
τ
, u,
u
˙
)
ð
ê❥❲✟❨❬❲
t
∈
R,
ǫ
∈
R
❙❱❩➯❛✯❩❬ò❇❛❜♠♥♠▲ï❉❛❜❨❬❛❜ò❇❲❸❪❫❲✟❨♦♣|
ǫ
|
< ǫ
0
,
τ
❙♥❩ó❛❑❨❭❲✌❛❜♠✮ï❥❛❜❨❫❛❜ò✙❲✟❪❬❲♦❨
❩❭❦❥❻❹ê✛❪❬ê❉❛q❪
|
τ
−
τ
0
|
< τ
1
í➐❣❜❨➯❩❬❣❳ò✙❲
0
< τ
1
<
τ
2
0
õ ②✛❣❜❨❬❲♦❣➠❯❜❲♦❨♦♣❉❪❬ê❥❲❵❻♦♠♥❣❜❩❬❲♦❤✫❣❳❨❭ö❥❙❱❪{(
u, v
)
∈
R
2
:
u
(
t
) =
u
0
(
t
)
, v
(
t
) = ˙
u
0
(
t
)
, t
∈
[0
, τ
0
]}
ö➳❲♦♠❱❣❳❘❥ú❳❩✲❪❬❣➽❪❬ê❥❲❵❨❬❲✟ú❳❙♥❣❳❘
{(
u, v
)
∈
R
2
:
u
2
+
v
2
< r
2
}
.
✐❢❘æ❪❫ê❥❲❵❩❬❛❜ò❇❲
ð
❛✌î✫❛❜❩✲í➐❣❳❨
(
L
)
♣❥❪❫ê❚❲ ➠❝❢❤❥❙❱ò❇❲✟❘❥❩❬❙❱❣❳❘❉❛❜♠▲❲♦ì❿❦❥❙❱❯q❛❜♠❱❲♦❘⑨❪ó❩❭î➅❩❞❪❫❲♦ò✦í➐❣❜❨(
LR
)
❙♥❩(
SL
)
x
˙
=
h
(
x
) +
ǫq
(
t
τ
, x
)
ð
ê❥❲✟❨❬❲
q
=
col
[
q
1
, q
2
]
,
q
1
=
−
γ
(
τ
t
, x
2
,
−
x
1
−
F
(
x
2
))
q
2
= 0
✡✁✭✌
✂✬✙ ✘ ✩✬✫✮✭ ✯✱✧✲✣✥✳✪✧ ✤✍✌ ☞ ✧ ✵✎✫✮✤ ✡ ✫✜✳ ✭✱✤✦✹✮✷ ✯✱✫✮✤✦✣✥✭ ✤✍✌
(
L
R
)
❡t❣
ð ð
❲
ð
❙♥♠♥♠✝❦❥❩❬❲❑❪❬ê❥❲➽❴✝❣❜❙♥❘❥❻♦❛❜❨✠✟❑ò✙❲✟❪❬ê❥❣➅❤☛í➐❣❳❨✇❪❫ê❚❲✯❤❥❲✟❪❬❲♦❨❬ò✙❙♥❘❥❛q❪❫❙❱❣❳❘☛❣❜í✑❪❫ê❚❲➽❛❜ï❥ï❥❨❭❣➠÷❿❝
❙❱ò❐❛q❪❬❲ ❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘☛❣❜í✗❪❫ê❚❲❑ï▲❲✟❨❭❪❬❦❥❨❬ö➳❲♦❤✕❲♦ì❿❦❉❛➷❪❫❙♥❣❜❘
(
LR
)
õþ➯ê❥❲✯❲✰÷❚❙❱❩❭❪❬❲♦❘❥❻✟❲✯❣❜í✑❪❫ê❥❲✣í➐❦❚❘❥❤❉❛➷❝
ò✙❲♦❘⑨❪❹❛q♠✵ò❇❛q❪❬❨❬❙➭÷✡❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘✛❣qíP❪❫ê❥❲✘✓❉❨❬❩❞❪✇❯q❛❜❨❭❙↔❛q❪❬❙♥❣❳❘❉❛q♠✮❩❭î➅❩❭❪❬❲♦ò✷❣❜í
˙
x
=
h
(
x
)
❛❜❘❚❤✛❪❫ê❚❲ ❦❥❘❚❙♥ì❿❦❥❲❵ï▲❲✟❨❬❙❱❣ø❤❥❙❱❻❷❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘p
(
t
)
❻✟❣❳❨❬❨❭❲♦❩❭ï▲❣❳❘❚❤❥❙♥❘❥ú✯❪❬❣u
0
(
t
)
❙❱❩ó❛❜❩❬❩❭❦❥ò❇❲✟❤
õ
✐❢❘❃❣❳❨❬❤❚❲♦❨✖❪❫❣✫ú❳❲✟❪■❲✟❩❭❪❫❙❱ò❐❛➷❪❫❲♦❩❷í➐❣❜❨❵❪❬ê❥❲➽❲❸÷ø❙♥❩❞❪❫❲✟❘❥❻♦❲❇❣❜íéï➳❲♦❨❭❙♥❣➅❤❥❙♥❻✯❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘❥❩
ð
❲✙ê❉❛✌❯❳❲➽❪❫❣
❻♦❛❜♠♥❻✟❦❥♠↔❛q❪❬❲➽❩❬❣❳ò✙❲❇❻✟❣❳❘❥❩❞❪❹❛❜❘⑨❪❫❩
õ
❺➋❣❜♠♥♠♥❣
ð
❙❱❘❥úæ❺➋❛❜❨✸✖➷❛❜❩✄✂❺
✄
☎ ♣✝❪❫ê❚❲❀ü❳❛❜❻✟❣❳ö❥❙❅ò❐❛➷❪❫❨❬❙➓÷
J
ê❉❛❜❩ ❪❬ê❥❲✖í➐❣❳♠♥♠❱❣ð
❙❱❘❥ú✣í➐❣❳❨❭ò
J
(
τ
0
) =
−
I
+
g
(
a
) 0
0
0
+
Y
(
τ
0
)
I
=
Id
2
❛❜❘❥❤
Y
(
t
)
❙♥❩➽❪❫ê❥❲✛í➐❦❥❘❥❤❉❛❜ò✙❲♦❘⑨❪❫❛❜♠ó❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘ ò❇❛q❪❬❨❬❙➭÷✺❣❜í❷❪❫ê❚❲✛❯q❛❜❨❫❛➷❪❫❙♥❣❜❘❉❛❜♠ ❩❞îø❩❞❪❫❲✟òð
❙❱❪❫ê
Y
(0) =
I
˙
y
=
O
g
′
(
u
0
(
t
))
−1
−
f
(
u
0
(
t
))
y.
✡ ⑦✭✌✐ô❪■❙♥❩✖ï❥❨❭❣➠❯❳❲♦❤✎♣✵❪❬ê❉❛q❪❵❙❱í
detJ
(
τ
0
)
6= 0
❪❫ê❥❲✟❘❃❪❬ê❥❲♦❨❭❲✙❲✰÷ø❙♥❩❭❪❵❦❚❘❥❙♥ì❿❦❥❲✟♠❱îù❤❥❲❸❪❫❲✟❨❬ò✙❙♥❘❥❲✟❤
í➐❦❥❘❚❻✟❪❫❙❱❣❳❘❥❩
τ
(
ǫ, φ
)
❛❜❘❚❤h
(
ǫ, φ
)
❤❥❲✔✓❥❘❥❲♦❤❇❙♥❘✯❪❫ê❥❲ó❘❥❲✟❙♥ú❳ê❿ö➳❣❳❨❬ê❚❣ø❣➅❤➽❣❜í(0
,
0)
❩❭❦❥❻❹ê❇❪❬ê❉❛q❪ ❪❬ê❥❲✖í➐❦❥❘❥❻✟❪❬❙♥❣❳❘u
(
t
;
φ, p
0
+
h
(
ǫ, φ
)
, ǫ, τ
(
ǫ, φ
))
❙❱❩❑❛✒ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻➽❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘ ❣❜íó❩❭î➅❩❭❪❬❲♦ò
(
SL
)
ð ❙❱❪❬êτ
(0
,
0) =
τ
0
❛❜❘❥❤
h
(0
,
0) = 0
õ ②✛❣❜❨❬❲♦❣➠❯❜❲♦❨♦♣P❛❜❘ ❲✟❩❭❪❬❙♥ò❇❛q❪❫❲➽❙♥❩■ú❳❙➭❯❳❲♦❘❃í➐❣❳❨✖❪❫ê❥❲✙❨❬❲✟ú❳❙♥❣❜❘ ❙❱❘ð
ê❥❙❱❻❹ê ❪❬ê❥❲❇❪❬ê❥❲✙❯q❛❜❨❭❙↔❛❜ö❚♠♥❲♦❩
ǫ
❛❜❘❚❤
φ
ò❇❛✌î✛❯q❛❜❨❞îõ
❺❉❣❳❨✇❲❸÷❚❛❜ò✙ï❥♠♥❲❑❙❱❘ù❲✟❯q❛❜♠❱❦❉❛q❪❫❙❱❘❥ú❐❪❬ê❥❲✯❘❥❣❳❨❭ò✸❣❜í❅❪❫ê❥❲✯❤❚❙❱ë▲❲♦❨❬❲✟❘❥❻♦❲✯❣qí
ü❳❛❜❻✟❣❳ö❥❙✩ò❇❛q❪❫❨❭❙♥❻✟❲♦❩
J
(
ǫ, φ, τ, h
)
−
J
(0
,
0
, τ
0
,
0)
.
s ❪❬❨❬❙❱❯➅❙♥❛❜♠
τ
0
❝ôï▲❲✟❨❬❙♥❣➅❤❥❙❱❻t❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘❐❣❜í ✡❖⑦✭✌➈❙♥❩
col
[
g
(
u
0
(
t
))
,
u
˙
0
(
t
)]
.
s ❪❬❨❬❙➭❯ø❙♥❛❜♠➋❻✌❛❜♠❱❻♦❦ø❝
♠♥❛q❪❫❙❱❣❳❘❀ú❳❙➭❯❳❲♦❩➯❪❬ê❥❲❵❣❜❪❫ê❥❲✟❨ó♠♥❙♥❘❚❲✌❛❜❨❭♠❱î✙❙♥❘❥❤❚❲♦ï➳❲♦❘❥❤❥❲✟❘⑨❪✇❩❭❣❳♠♥❦❚❪❬❙♥❣❳❘✡❣qí ✡❖⑦✭✌
col
[
g
(
u
0
(
t
))
v
(
t
)
,
u
˙
0(
t
)
v
(
t
) +
g
(
u
0
(
t
))
g
′
(
u
0
(
t
))
˙
v
(
t
)]
ð
ê❥❲✟❨❬❲
v
(
t
) =
Z
t
0
[
g
0
(
s
)]
−
2
g
′
(
u
0
(
t
))
exp
[−
Z
s
0
f
(
u
0
(
σ
))
dσ
]
ds
í➐❣❳❨
t
∈
[0
, τ
0
]
õþ➯ê❥❲✟❘✛❪❫ê❚❲❵í➐❦❚❘❥❤❉❛❜ò✙❲♦❘⑨❪❹❛q♠✮❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘✡ò❇❛q❪❫❨❭❙➭÷❐❣❜í ✡❖⑦✭✌
ð
❙❱❪❬ê
Y
(0) =
I
❙♥❩Y
(
t
) =
g
(
u
0
(
t
))
g
(
a
)
g
(
a
)
g
(
u
0
(
t
))
v
(
t
)
˙
u
0
(
t
)
g
(
a
)
g
(
a
) ˙
u
0
(
t
)
v
(
t
) +
g
(
a
)
g
(
u
0
(
t
))
g
′
(
u
0
(
t
))
v
˙
(
t
)
.
s➯❻✟❻♦❣❳❨❭❤❥❙♥❘❥ú ❪❫❣ r✩❙♥❣❜❦❚❯➅❙♥♠♥♠❱❲ ❩✡í➐❣❳❨❬ò❑❦❚♠↔❛ ❪❬ê❥❲
✎
❨❬❣❜❘❥❩✸✖❿❙↔❛q❘④❤❥❲✟❪❬❲♦❨❬ò✙❙♥❘❥❛❜❘⑨❪
W
(
t
)
ð ❙❱❪❬êW
(
o
) = 1
❙❱❩➯ú❳❙❱❯❜❲♦❘æö⑨îW
(
t
) =
exp
[−
Z
t
0
f
(
u
0
(
τ
))
dτ
]
.
þ➯ê❥❲❵❻❹ê❉❛q❨❫❛❜❻❸❪❫❲♦❨❭❙♥❩❞❪❫❙♥❻❵ò❑❦❚♠❱❪❫❙❱ï❥♠♥❙❱❲♦❨❭❩➈❣❜í ✡ ⑦ ✌✲❛❜❨❬❲
ρ
1
= 1
❛q❘❥❤
ρ
2
=
W
(
τ
0
) =
exp
[−
Z
τ
0
0
f
(
u
0
(
τ
))
dτ
]
.
ρ
2
<
1
❙➭í✝❛❜❘❥❤✫❣❳❘❥♠❱î❐❙❱í
Z
τ
0
0
f
(
u
0
(
τ
))
dτ >
0
.
✡
✄
⑩✭✌
þ➯ê❥❲■❙♥❘❥❙➭❪❫❙♥❛❜♠➋❻♦❣❳❘❥❤❚❙❱❪❫❙❱❣❳❘❥❩➯ú❳❙➭❯❳❲
Y
(
τ
0
) =
1
g
2
(
a
)
v
(
τ
0
)
0
ρ
2
.
þ➯ê❿❦❥❩✟♣
ð
❲■ú❳❲✟❪
J
=
g
(
a
)
g
2
(
a
)
v
(
τ
0
)
0
ρ
2
−
1
,
J
−
1
=
g
−
1
(
a
)
g
2
(
a
)
v
(
τ
0
)(1
−
ρ
2
)
−
1
0
−(1
−
ρ
2
)
−
1
.
þ➯ê❥❲✟❨❬❲❸í➐❣❳❨❬❲q♣
||
J
−
1
||= 2
max
[
g
−
1
(
a
)
,
(1
−
ρ
2
)
−
1
, g
2
(
a
)
v
(
τ
0
)(1
−
ρ
2
)
−
1
]
.
þ➯ê❥❲■❙♥❘⑨❯❳❲✟❨❬❩❭❲ ò❇❛q❪❫❨❭❙➭÷❇❣❜í
Y
(
t
)
❙♥❩Y
−
1
(
t
) =
W
(
t
)
g
(
a
) ˙
u
0
(
t
)
v
(
t
) +
g
(
a
)
g
g
′
(
(
u
u
0
0
(
(
t
t
))
))
v
˙
(
t
)
−
g
(
a
)
g
(
u
0
(
t
))
v
(
t
)
−
u
˙
0
(
t
)
g
(
a
)
g
(
u
0
(
t
))
g
(
a
)
.
❡☎❣
ð ð
❲✖ê❉❛✌❯❳❲❷❪❫❣✣❤❥❲✟❪❬❲♦❨❬ò✙❙♥❘❚❲☎❪❫ê❚❲❷❻♦❣❳❘❚❩❭❪❹❛q❘❿❪❬❩✲í➐❣❳❨é❩❞î➅❩❭❪❫❲✟ò
(
LR
)
õ ❺❉❣❳♠❱♠♥❣ð
❙♥❘❥ú ✂❺
✄
☎
♠❱❲✟❪ó❦❥❩ó❤❥❲✟❘❥❣❜❪❫❲
S
=
{
x
= (
x
1
, x
2
)
∈
R
2
/ x
2
2
+ [−
x
1
−
F
(
x
2
(
t
))]
2
< r
2
}
g
0
:=
maxx
∈
S
|
g
(
x
2
)
|
,
g
1
:=
maxx
∈
S
|
g
′
(
x
2
)
|
,
g
2
=
maxx
∈
S
|
g
′′
(
x
2
)
|
.
✡
✄✂✄
✌
f
1
:=
maxx
∈
S
|
f
(
x
2
)
|
,
f
2
:=
maxx
∈
S
|
f
′
(
x
2
)
|
q
0
:=
maxx
∈
S,s
∈
R
|
q
(
s, x
)
|
,
q
1
:=
maxx
∈
S,s
∈
R
|
q
x
′
(
s, x
)
|
,
q
2
:=
maxx
∈
S,s
∈
R
|
q
s
′
(
s, x
)
|
.
✡
✄
✌
K
:=
max
t
∈
[
−
τ
0
2
,τ
0
]
|
Y
(
t
)
|
,
K
−
1
:=
max
t
∈
[
−
τ
0
2
,τ
0
]
|
Y
−
1
(
t
)
|
.
þ➯ê❿❦❥❩✟♣
ð
❲■ò❐❛✌î❀❤❚❲♦❤❥❦❥❻✟❲■❪❬ê❉❛q❪
P
:=
max
t
∈
[
−
τ
0
2
,τ
0
]
|
p
˙
(
t
)
|≤
K
2
.
þ➯ê❥❲❵❙❱❘❥❙❱❪❬❙↔❛❜♠➳ï❥ê❉❛❜❩❭❲
φ
❛q❘❥❤✡❪❫ê❥❲❵ï➳❲♦❨❬❙❱❣ø❤τ
ê❉❛✌❯❳❲✖❪❬❣✙❯❜❲♦❨❬❙➭í î❀❪❫ê❚❲❷í➐❣❳♠♥♠❱❣ð
❙❱❘❥ú❥♣
ð
ê❥❙❱❻❹ê✛❻♦❛❜❘
ö➳❲❵❲✌❛❜❩❭❙♥♠❱î❇❣❳ö❚❪❫❛❜❙♥❘❚❲♦❤✡í➐❨❬❣❳ò ❪❫ê❥❲■❛❜ö➳❣➠❯❳❲■❲♦❩❞❪❫❙♥ò❇❛q❪❬❲♦❩♦♣❚❩❬❲♦❲ ✂❺
✄
☎
φ <
τ
0
2
,
|
τ
−
τ
0
|
<
τ
0
2
.
✐ôí✝❙♥❘✫❛❜❤❥❤❚❙❱❪❫❙❱❣❳❘
ð
❲■❩❭❦❥ï❥ï➳❣❳❩❬❲
ǫ
❛❜❘❥❤h
❛❜❨❭❲■❩❭❦❥❻❹ê✛❪❬ê❉❛q❪3
2
g
0
|
ǫ
|
+
|
h
|
< σexp
(−
3
2
g
1
τ
0
)
✡➐ê❥❲♦❨❭❲
σ
❙♥❩❑❪❬ê❥❲æ❤❥❙♥❩❞❪❹❛❜❘❚❻♦❲æö▲❲❸❪ð
❲✟❲♦❘ ❪❬ê❥❲æï❉❛q❪❬ê ❣❜ít❪❫ê❚❲✫ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻❐❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘ ❛❜❘❥❤ ❪❫ê❥❲
ö➳❣❳❦❥❘❥❤❥❛❜❨❭î✡❣qí
S
✌➈❪❫ê❚❲♦❘✛❛➽❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘✡❣❜í(
LR
)
❲❸÷ø❙♥❩❞❪❫❩õ
þ➯ê❥❲✟❘✮♣
ð
❲❵ê❉❛✌❯❜❲ ï❥❨❭❣➠❯❳❲♦❤✫❪❫ê❥❲✖í➐❣❳♠❱♠♥❣
ð
❙♥❘❚ú
✂✁☎✄✝✆✟✞✠✄☛✡ ☞ ✌✎✍
1
✵✑✏ ✪✒✏✴✵✔✓ ✱ ★✒✳ ✙✖✕ ✪✗✦ ✪ ✙✴✤ ✳✴✦ ✵✑✏✴✤✵✶✙✗✓✙✘ ★✰✤✧✵✱ ★✰✵✶✳✴✦ ✛ ✍✛✚✢✜✤✣✥✚ ✤✎✕ ✪✗✤ ✓ ✳✸✪✗✢✦✏ ✵✩✢ ✳★✧✩✘ ✪✗★✰✵✩✤✧✯ ✚✢✪☛✫✖✣ ✕✥✛✣★✫✷✠✏ ✣ ✤✎✕✥✳✴✢ ✤✎✕✥✳✴✦ ✳✚✪✗✦ ✳ ✤✭✬ ✛ ✍ ✘ ✢ ✙✴✤✵✶✛✣✢✦✏τ, h
:
U
→
R
✪✗✢ ✷ ✪ ✙✜✛✣✢✦✏✴✤ ✪✗✢✥✤τ
1
<
τ
2
0
✏✤✘✍✙✖✕ ✤✎✕ ✪✗✤ ✤✔✕✥✳✲✱✥✳✴✦✴✵✶✛✮✷✗✵✶✙✮✏ ✛✣★✯✘ ✤ ✵✶✛✣✢
u
(
t, φ, a
+
h
(
ǫ, φ
)
, ǫ, τ
(
ǫ
))
✛ ✍ ✳★✧✩✘ ✪✗✤✧✵✶✛✣✢
(
LR
)
u
¨
+
f
(
u
) ˙
u
+
g
(
u
) =
ǫγ
(
t
τ
(
ǫ
)
, u,
u
˙
)
✳✱✰ ✵✑✏✴✤✔✏ ✍ ✛✣✦
(
ǫ, φ
)
∈
U
✲ ✪✗✢ ✷|
τ
−
τ
0
|
< τ
1
, τ
(0
,
0) =
τ
0
, h
(0
,
0) = 0
.
s☎❩P❛☎❻✟❣❳❨❬❣❜♠♥♠↔❛q❨❭î
ð
❲✲ò❇❛➠î❵❤❚❲♦❤❥❦❥❻✟❲✲í➐❨❬❣❳ò✿❪❬ê❥❲✲❛qö▲❣➠❯❳❲é❩❬❣❳ò✙❲➈❨❭❲♦❩❭❦❥♠❱❪❬❩P❛❜ö➳❣❳❦❚❪✬❛❜❦❚❪❬❣❳❘❥❣❳ò✙❣❳❦❥❩
ï➳❲♦❨❞❪❫❦❥❨❭ö❉❛q❪❫❙❱❣❳❘❥❩➯❣❜í✵❪❫ê❚❲ r✩❙❱❲♦❘❉❛❜❨❭❤æ❩❭î➅❩❭❪❬❲♦ò
(
LR
A
)
u
¨
+
f
(
u
) ˙
u
+
g
(
u
) =
ǫγ
ð
ê❥❲✟❨❬❲ó❪❫ê❥❲tï➳❲♦❨❭❪❬❦❥❨❬ö❥❛q❪❫❙❱❣❳❘❇❙❱❩✑❙♥❘❥❤❥❲✟ï▲❲✟❘❥❤❥❲✟❘❿❪➈❣❳❘✙❪❫ê❥❲ó❪❫❙❱ò❇❲✲❯q❛❜❨❭❙↔❛❜ö❥♠❱❲
γ
≡
γ
(
u,
u, ǫ, τ
˙
)
õ þ➯ê❥❲❵❲✟ì➅❦❚❙❱❯q❛❜♠♥❲✟❘⑨❪☎ï❚♠↔❛❜❘❥❲✖❩❞î➅❩❭❪❫❲✟ò ❙♥❩➯❣❜í✵❪❫ê❚❲❵í➐❣❜❨❬ò˙
x
=
h
(
x
) +
ǫq
(
t
τ
, x
)
ð
ê❥❲✟❨❬❲
q
=
col
[
q
1
, q
2
]
.
✉é❣❳❘❚❩❬❙♥❤❚❲♦❨➯❪❫ê❥❲✖í➐❣❳♠❱♠♥❣
ð
❙♥❘❚ú❑❛❜❦❚❪❬❣❳❘❥❣❳ò✙❣❳❦❥❩➯ï➳❲♦❨❞❪❫❦❥❨❭ö▲❲✟❤✒r✩❙❱❲♦❘❉❛❜❨❭❤æ❲♦ì❿❦❉❛q❪❬❙♥❣❳❘✫❣❜í✵❪❫ê❥❲✖í➐❣❳❨❭ò
(
LR
A
)
u
¨
+
f
(
u
) ˙
u
+
g
(
u
) =
ǫγ
≡
γ
(
u,
u, ǫ, τ
˙
)
ð
ê❥❲✟❨❬❲
t
∈
R,
ǫ
∈
R
❙❱❩➯❛✯❩❬ò❇❛❜♠♥♠▲ï❉❛❜❨❬❛❜ò❇❲❸❪❫❲✟❨♦♣|
ǫ
|
< ǫ
0
,
τ
❙♥❩ó❛❑❨❭❲✌❛❜♠✮ï❥❛❜❨❫❛❜ò✙❲✟❪❬❲♦❨
❩❭❦❥❻❹ê✛❪❬ê❉❛q❪
|
τ
−
τ
0
|
< τ
1
í➐❣❜❨➯❩❬❣❳ò✙❲
0
< τ
1
<
τ
2
0
õ ②✛❣❜❨❬❲♦❣➠❯❜❲♦❨♦♣❉❪❬ê❥❲❵❻♦♠♥❣❜❩❬❲♦❤✫❣❳❨❭ö❥❙❱❪{(
u, v
)
∈
R
2
:
u
(
t
) =
u
0
(
t
)
, v
(
t
) = ˙
u
0
(
t
)
, t
∈
[0
, τ
0
]}
ö➳❲♦♠❱❣❳❘❥ú❳❩✇❪❬❣❀❪❫ê❥❲✯❨❭❲♦ú❳❙❱❣❳❘
S
=
{(
u, v
)
∈
R
2
:
u
2
+
v
2
< r
2
}
. γ
❙❱❩❵❛❐í➐❦❥❘❥❻❸❪❫❙♥❣❜❘ù❣❜í ❻✟♠↔❛❜❩❭❩C
2
.
✐❢❘æ❪❫ê❥❙❱❩➯❻✌❛❜❩❭❲✖❪❫ê❥❲❵ï➳❲♦❨❭❪❬❦❥❨❬ö❥❛q❪❫❙❱❣❳❘æ❙♥❩➯❙♥❘❚❤❥❲♦ï➳❲♦❘❥❤❚❲♦❘⑨❪✇❣❜❘æ❪❬ê❥❲❵❙♥❘❥❙➭❪❫❙♥❛❜♠▲ï❚ê❉❛❜❩❬❲
φ
õ✎
❲✖❪❫ê❥❲✟❘✛ê❉❛✌❯❜❲■❪❬ê❥❲✖í➐❣❳♠♥♠❱❣
ð
❙❱❘❥ú
✁
✆✟✞✠✆✄✂☎✂☎✆✦✞✞✝✠✟ ✡
✘ ✱✭✱✥✛ ✏ ✳✚✵✩✢ ✳★✧✩✘ ✪✗★✰✵✩✤✯
✚✢✪☛✫✖✣ ✕✥✛✣★✫✷✠✏
✲
✤✎✕✥✳✴✢ ✤✔✕✥✳✴✦✜✳ ✪✗✦ ✳ ✙✜✛✣✢✦✏✴✤✶✪✗✢✥✤✔✏
ǫ
0
✪✗✢ ✷
τ
1
<
τ
2
0
✏✤✘✥✙✖✕ ✤✎✕ ✪✗✤ ✤✁✛✺✳✸✪ ✙✖✕
ǫ
∈
[−
ǫ
0
, ǫ
0
]
✤✔✕✥✳✴✦ ✳✝✳✱✰ ✵✑✏✴✤ ✤✭✬ ✛ ✍ ✘ ✢ ✙✴✤✧✵✶✛✣✢✦✏
τ, h
✛✣✢✥★✰✯✕✷ ✳ ✱✥✳✴✢ ✷ ✳✴✢✥✤ ✛✣✢ǫ
:
τ
≡
τ
(
ǫ
)
, h
≡
h
(
ǫ
)
✏✤✘✥✙✖✕ ✤✎✕ ✪✗✤✲✤✔✕✥✳ ✱ ✦✜✛✗✬✮★✒✳✤✓LR
A
)
u
¨
+
f
(
u
) ˙
u
+
g
(
u
) =
ǫγ
≡
ǫγ
(
u,
u, ǫ, τ
˙
)
u
(0) =
a
+
h
(
ǫ
)
,
u
˙
(0) = 0
✡
✄
⑥✭✌
✕ ✪✠✏ ✪ ✘ ✢✥✵✭✧✩✘✍✳✄✱✥✳✴✦ ✵✶✛✮✷✗✵✶✙ ✢ ✛✣✢ ✙✜✛✣✢✦✏✴✤✶✪✗✢✥✤ ✏ ✛✣★✯✘ ✤ ✵✶✛✣✢
u
(
t, ǫ
)
✬ ✵✩✤✔✕ ✱✥✳✴✦✴✵✶✛✮✷τ
(
ǫ
)
☛ ☞ ✛✣✦ ✳✜✛✍✌✗✳✴✦✲
τ
(0) =
τ
0
,
✪✗✢ ✷|
τ
−
τ
0
|
< τ
1
, τ
(0) =
τ
0
, h
(0) = 0
.
✞ ❘✕❣❜❪❫ê❥❲✟❨✖ê❉❛❜❘❥❤✎♣●❛❀❩❬❲♦❻✟❣❳❘❥❤ ❩❬ï➳❲♦❻♦❙♥❛❜♠✬❻♦❛❜❩❬❲➽ò❇❛✌î✒❣➅❻♦❻✟❦❥❨
ð
ê❥❲♦❘✕❪❫ê❚❲✙ï➳❲♦❨❞❪❫❦❥❨❭ö❉❛q❪❬❙♥❣❳❘
❤❥❣➅❲♦❩é❘❥❣❜❪✲❤❚❲♦ï➳❲♦❘❥❤æ❣❳❘❐❪❬ê❥❲❷❩❞❪❹❛q❪❬❲❷❣❜í✩❪❫ê❚❲❷❩❭î➅❩❭❪❬❲♦ò
õ
þ➯ê❉❛q❪➯ò✙❲✌❛❜❘❚❩➈❪❫ê❚❲❷ï▲❲✟❨❭❪❬❦❥❨❬ö❉❛➷❪❫❙♥❣❜❘❀❙♥❩
❙❱❘❥❤❥❲♦ï➳❲♦❘❚❤❥❲♦❘⑨❪t❣❳❘
u
γ
≡
γ
(
t
τ
)
.
õ ✂✁☎✄✆✁☎✂✁☎✝✟✞✠✁☛✡ ✂✠❅❝➦❺ ☎ ❴ õ ✠é❦ø❪❬❪❹❛ ✑✮✑♦❣❜❘❥❙✲❛❜❘❚❤➩s õ ❺❉❣❳❘❥❤❉❛ ☞✆✳✴✦ ✵✶✛✮✷✗✵✶✙ ✱✥✳✴✦ ✤ ✘ ✦✔✬✜✪✗✤ ✵✶✛✣✢✦✏ ✛ ✍ ✏ ✙✸✪✗★✫✪✗✦ ✏ ✳✜✙✜✛✣✢ ✷✝✛✣✦ ✷ ✳✴✦✚✷✗✵✌ ✳✴✦ ✳✴✢✥✤ ✵✧✪✗★ ✳★✧✩✘ ✪✗✤ ✵✶✛✣✢✦✏ ÿ☎❙♥❩❬❻✟❨ õ ❛q❘❥❤ù✉é❣❳❘⑨❪ õ ÿ☎î➅❘ õ ①➅î➅❩❭❪ õ ♣➋❯❳❣❜♠✝⑥ø♣➳❘
◦
⑥❚♣❚ï õ ⑧❿⑤ ✄ ❝ô⑧❿⑤❳⑤ø♣ ✡ ✄ ⑦❜⑦⑨③ ✌ õ ✂✉❅❝❞✉ ☎ ✈ õ ✉éê❚❣❳❦❥❙✒✖❿ê❉❛✣❛❜❘❥❤æ❺ õ ✉é❦❚❯❳❲✟♠♥❙❱❲♦❨ ✍ ✳✤✓✄✪✗✦✏✎ ✏✘✛✣✢ ✏ ✛✠✓ ✳ ✓ ✛✣✢ ✛✣✤✁✛✣✢✥✵✶✙✴✵✩✤✧✯✙✜✛✣✢ ✷✗✵✩✤✧✵✶✛✣✢✦✏ ✍ ✛✣✦ ✤✎✕✥✳ ✱✥✳✴✦✴✵✶✛✮✷ ✍ ✘ ✢ ✙✴✤✧✵✶✛✣✢ s☎ï❥ï❥♠
õ ②✒❛➷❪❫ê õ ✝✣♣❳❘❥❣ õ ⑥❚♣ ➷⑧⑨⑥✒✑ ❳⑤ ø♣✥✡ ✄ ⑦❳⑦❳⑦✭✌ õ ✂✉❅❝✁ ☎ ① õ ❡ õ ✉éê❥❣ ð ❝❞ü õ
✇❛❜♠❱❲ ☞ ✳✴✤✎✕✥✛✮✷✠✏ ✛ ✍ ✬✮✵✍ ✘ ✦ ✙✸✪✗✤ ✵✶✛✣✢ ✤✔✕✥✳✜✛✣✦ ✯q♣ ①øï❥❨❬❙❱❘❥ú❳❲✟❨♦♣
✠é❲✟❨❬♠❱❙♥❘ ✡
✄
⑦ ✌
õ
✂✉❅❝➦r✻☎ û
õ
✉é❣ø❤❚❤❥❙♥❘❥úq❪❫❣❳❘ø❝❢❡
õ
r✩❲✟❯➅❙♥❘❥❩❭❣❳❘ ✓ ✕✥✳✜✛✣✦ ✯ ✛ ✍ ✛✣✦ ✷✗✵✩✢ ✪✗✦ ✯ ✷✗✵✌ ✳✴✦ ✳✴✢✥✤✧✵✧✪✗★ ✳★✧✩✘ ✪✕✔
✤✧✵✶✛✣✢✦✏❸♣ ②✫❻✣❧✖❨❫❛ ð ❝ ☎❙♥♠❱♠ ♣➅❡☎❲ ð ❝✗✖❅❣❳❨ ✖➋♣ ✡ ✄ ⑦⑨⑤❜⑤ ✌ õ ✂❺ ✄ ☎ ② õ ❺➋❛❜❨ ✖➷❛❜❩ ✘ ✏✴✤✧✵✔✓✄✪✗✤ ✳✖✏ ✛✣✢ ✤✎✕✥✳ ✳✱✰ ✵✑✏✴✤✁✳✴✢ ✙✜✳✚✦ ✳✗✙✗✵✶✛✣✢✦✏✄✛ ✍ ✱✥✳✴✦ ✤✭✘ ✦✔✬✴✳✸✷ ✱✥✳✚✔
✦ ✵✶✛✮✷✗✵✶✙ ✏ ✛✣★✯✘ ✤ ✵✶✛✣✢✦✏❸♣ ①ø✐❢s✖②✴ü
õ ②✒❛q❪❬ê õ s✇❘❉❛❜♠ õ ♣ø❯❳❣❳♠●⑦❚♣❚ï õ ✝⑨③✻❝ ❳⑦❳⑩❚♣ ✡ ✄ ⑦❳③ ✭✌ õ ✂❺ ✣☎ ② õ ❺➋❛❜❨ ✖➷❛❜❩ ☞✆✳✴✦✴✵✶✛✮✷✗✵✶✙ ✓ ✛✣✤✧✵✶✛✣✢✦✏ ☛ s☎ï❥ï❥♠❱❙♥❲♦❤④②✒❛q❪❫ê❚❲♦ò❇❛q❪❫❙❱❻✌❛❜♠ ①ø❻✟❙➭❝ ❲✟❘❥❻♦❲✟❩ õ☎✄ ⑩❜⑧ õ ❡t❲ ð ✖❅❣❜❨✸✖➋♣➋❡✛✖❑♣➳①øï❥❨❬❙❱❘❥ú❳❲✟❨❞❝ô❶✑❲✟❨❬♠♥❛❜ú õ ✡ ✄ ⑦❳⑦❜⑧✭✌ õ
✂❺✵❝❢s ☎ ②
õ
❺➋❛❜❨✸✖➷❛❜❩➦❝❢s☎ö▲❤❥❲✟♠ ■❛q❨❬❙♥ò ✜ ✢ ✙✜✛✣✢✥✤ ✦ ✛✣★✩★✫✪✭✬✮★✰✯ ✱✥✳✴✦✴✵✶✛✮✷✗✵✶✙ ✱✥✳✴✦ ✤✭✘ ✦✔✬✜✪✗✤✧✵✶✛✣✢✦✏
✛ ✍✣✢ ✵✶✳✥✤✢ ✪✗✦✸✷✦✤✏✄✳★✧✩✘ ✪✗✤ ✵✶✛✣✢ ❴✝❲✟❨❬❙♥❣➅❤❥❙❱❻✌❛❀❴✵❣❳♠❱î❿❪❫❲✟❻❹ê õ û✑♠❱❲♦❻ õ û✑❘❚ú❳❙♥❘❥❲✟❲♦❨❭❙♥❘❥ú❥♣ ✄ ✝❐ï õ ⑧ ✄ ❝❢⑧❿⑤ ✡ ✄ ⑦⑨③✂ ✌ õ ✂✇❛ ☎ ü õ ☎❛❜♠♥❲ ✓ ✛✜✱ ✵✶✙✖✏ ✵✩✢ ✷✗✯ ✢ ✪ ✓ ✵✶✙ ✬✮✵✍ ✘ ✦ ✙✸✪✗✤ ✵✶✛✣✢ ✤✎✕✥✳✜✛✣✦ ✯➷♣ ✎ ❙❱♠♥❲❸î❳♣➈❡☎❲ ð ❝ ✖❅❣❳❨ ✖➋♣ ✡ ✄ ⑦ ✄ ✌ õ ✂ ☎ ❴ õ ✇❛❜❨❞❪❫ò❇❛❜❘ ✜ ✢ ✬✴✛✠✘ ✢ ✷✭✪✗✦ ✯ ✌✣✪✗★✯✘✍✳ ✱ ✦ ✛✗✬✮★✒✳✤✓✂✏ ✍ ✛✣✦✮✏✤✘ ✱✥✳✴✦ ★✰✵✩✢ ✳✸✪✗✦✮✏ ✳✜✙✜✛✣✢ ✷ ✛✣✦ ✷ ✳✴✦✘✷✗✵✌ ✳✴✦ ✳✴✢✥✤ ✵✧✪✗★ ✳★✧✩✘ ✪✗✤✧✵✶✛✣✢ ü õ ❣qíPÿ✇❙➭ë✕û✗ì õ ♣❥❯❜❣❳♠ ✝❚♣❚ï õ ⑥⑨③✻❝❞⑤q⑥❚♣ ✡ ✄ ⑦⑨③❳③ ✌ õ
✂r●❣ ☎
✎ õ
①
õ
r✩❣❳❦❥❤ ☞ ✳✴✦ ✵✶✛✮✷✗✵✶✙✗✏ ✛✣★✯✘ ✤ ✵✶✛✣✢✦✏ ✛ ✍
x
′′
+
cx
′
+
g
(
x
) =
ǫf
(
t
)
②✛❲✟ò õ s☎ò❇❲✟❨ õ ②✒❛➷❪❫ê õ ①ø❣➅❻ õ ♣❉❘✫⑥ ✄ ♣❥ï õ ✄ ❝➦⑤❳③ø♣✻✡ ✄ ⑦⑨⑤q⑦✭✌ õ
