Directory UMM :Journals:Journal_of_mathematics:EJQTDE:

✂✁☎✄✝✆✟✞✡✠☛✆✌☞ ✂✁☎✄✎✍✑✏✒✄✝✓✕✔✖✍✗✆✟✞✙✘✛✚✜✞✣✢✥✤✦✞✙✘✡✧✩★✪✞✙✘✫✚✬✁☎✄✮✭✯✔✖✍✗✆✰✭✙✁
✱ ✁☎☞✲✞✙✘☛✠ ✳✴✄✵✠✛✁☎✄✷✶✸✆✰✹✺✁☎✄✩✁☎✘✙✍✑✆✻✔✣✼✂✽✿✾❀✏✕✔✖✍✗✆✟✞✙✘✛✚
❁❃❂❅❄❇❆❉❈✎❊●❋■❍❑❏✝❈▲❊✬▼❖◆✩❏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❢❽➋➊ ➀⑨➌⑨➁➃➍q➎➐➏⑨➑❢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✌➺❫➫➃➀⑨➻➽➉❢➁➓➼✇❾✌➌❿➾✌➒✑➚❢➪♦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❢Ï↔➀⑨Ð➦✃➠➑❢➵❥Ñ➋➤❹Ò➅➢❷Ó❿Ô♥➒❢Õ ➧❳➺ ➒❢Ð❹➉➦❒➯➤❫➇❵Ö✎Ñ×➒✰✃ ➘✌➶❞➥❉➶ Ï➁➓Ø✝➤❫➌➅Ð➦✃✻➄♦❒↔➑➦Ï ➢❷➾✌Ù❐➤❹➴➷Ú ➀➅➄♦→♦➉❢Û ➁➓❾✌➡P➌➋Ü➠➘✌Ý ➏❥➘➅❾✌→♦➫➃➧❜Û ➌❿➡ ❾✌→ ➇✖Ý ➁➭➄♦➫❳➒➦➧❜➒❢➉➦➤❫➇❵➒❸➊
′′

′

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❇❙❱❘❥❛❜❤❥♠♥♠ó❲♦ï➳ï❉❲♦❛q❘❚❨❫❤❥❛❜ò✙❲♦❘⑨❲✟❪t❪❬❲♦❣❜❨♦í ♣ g ❙♥❩❀❛ T ❝ôï▲❲✟❨❬❙♥❣➅❤❥❙❱❻✫í➐❦❥❘❚❻✟❪❫❙❱❣❳❘✂❙❱❘ t ❛q❘❥❤
g(t, x, 0, 0) = g(x)
✐❢❘❀❪❬ê❥❲✖❻✌❛❜❩❭❲ ð ê❥❲♦❨❭❲ g ❙❱❩✲❙♥❘❚❤❥❲♦ï➳❲♦❘❥❤❚❲♦❘⑨❪ó❣❜tí õ x ❛❜❘❥❤✡❙❱❩✲❻♦❣❳❘⑨❪❬❙♥❘❿❦❥❣❳❦❥❩❭♠❱î❐❤❥❙❱ë▲❲♦❨❭❲♦❘⑨❪❫❙♥❛❜ö❥♠♥❲t❙♥❘
❪❬ê❥❲■❲❸÷ø❙♥❩❞❪❫❲♦❘❚❻♦❲❑ï❥❨❭❣❳ö❥♠❱❲♦ò✦❣❜íP❘❥❣❳❘✒❻✟❣❳❘❥❩❞❪❹❛❜❘⑨❪❷ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻✖❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘❥❩ó❣❜í (E ) ê❉❛q❩☎ö➳❲♦❲✟❘
x
❩❞❪❫❦❥❤❥❙❱❲♦❤✫ö⑨î✡ò❐❛q❘❿î✡❛❜❦❚❪❫ê❚❣❳❨❬❩ õ
✐❢❘❥❤❚❲♦❲♦❤✎♣✩❙♥❘✛❪❬ê❥❲❑♠♥❛q❪❬❪❬❲♦❨✇❻♦❛❜❩❬❲✣❻♦❲♦❨❞❪❹❛❜❙❱❘ù❛❜ò✙❣❳❘❥ú✙❪❫ê❥❲✟ò✸ï❥❨❭❣✻❯❜❲♦❤✕❲❸÷ø❙❱❩❭❪❫❲✟❘❥❻♦❲✯❣qí✑❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘❥❩
û➈ü⑨ý❷þóÿ❷û➯♣✁❜⑩❳✂⑩ ➽❡t❣ õ ⑧❥♣❥ï õ☎✄
′

ǫ

❜❣ 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➳❲
✁

′′

✠

☛✡

✄✂

☞

✁✆☎

✝

✞✂✟✆☎

✍✌

′

✒✑

✏✎

t
x′′ + f (x)x′ + g(x) = ǫh( , x, x′ , ǫ)
T

(1ǫ )

ð❙❱❩❭íê❥î➅❲✟❙♥❨❬❘❥❲ ú✫h❻♦❣❜❙♥❩❘❥❤❥T❙❱❪❬❝❢ï➳❙♥❣❳❲♦❘❚❨❬❩■❙❱❣ø❤❚❤❚❲ ❙♥❻❵❉❘❥❙♥❘ ❲✟❤ñt ♣ ö➳f❲♦♠♥❣ ❛❜❘❚❤ g ❛❜❲❇❨❬❲■♠❱❣øí➐❣ ❦❥✒❘❥❻✟í➐❣❳❪❬❙♥❨■❣❳❘❚ï▲❩✇❲✟❨❬❣❳❙♥❣➅❘❥❤❥♠➭î✫❙❱❻✯❤❥❩❬❲♦❣❳ï➳♠❱❲♦❦❚❘❥❪❫❤❚❙❱❣❳❲♦❘❥❘⑨❩❵❪❵❣❜❣❳í ❘ x ♣✮❷❩❬❛qí➐❣❳❪❭❨❝
❩ǫ ❬ò❇❛❜♠♥♠➈❲♦❘❥❣❜❦❥ú❳ê ❦❚❘❥❤❥❲♦❨➽❩❭❣❳ò❇❲✫❛❜❤❚ð ❤❥õ ❙❱❪❬❙♥❣❳❘❉❛q♠éê⑨î➅ï▲❣❜❪❬ê❥❲♦❩❭❙♥❩ õ ✐ô❪✙❙♥❩➽❛q❩❬❩❬❦❚ò❇❲✟❤ ❪❫ê❥1❛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✫❩❬ê❉❛✯❛❜ï▲♠❱♠✮❲✟ï❥❨❬❙♥❨❭❣➅❣✻❤ ❯❜❲✣ð ❛qê❥❘✫❙♥❻❹❲✰ê✫÷ø❻✌❙♥❩❭❛q❪❬❘✫❲♦❘❥ö➳❻✟❲❵❲■❻❹❪❬ê❥ê❥❣❳❲♦❩❭❣❜❲♦❨❬❘☛❲♦ò✜❛qï❥í➐ï❥❣❳❨❬❨✲❣❜❪❫ï❥ê❥❨❬❙❱❙♥❩ó❛q❪❫❲♦❲✟ì❿♠❱❦❉î ❛qõ ❪❬❙♥❣❳❘
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❪❬❙♥❣❳❘
✔✓

✗✖

✕✎

✘✡

✌

ǫ

✚✙✜✛✣✢✥✤✧✦✜✛✣★✩★✫✪✭✬✮★✰✯✲✱✥✳✴✦✴✵✶✛✮✷✗✵✶✙

✸✖

✹✂

✣☎

✎

✺✂ ✻☎

x′′ + cx′ + g(x) = ǫh(t),

✡

✄

✌

ð❛➩ê❥❯q❲✟❛❜❨❬❨❭❲✕❙↔❛❜❪❫❘⑨ê❥❪✡❲❃❣❜ï➳í❵❲♦❪❬❨❭ê❥❪❬❲ù❦❥❨❬❙♥ö❥ò✙❛q❪❫ï❥❙❱♠♥❣❳❙❱❘❻♦❙➭❪✙❤❥í➐❣ø❦❚❲✟❘❥❩✛❻✟❪❬❘❚❙♥❣❜❣❳❪✫❘✂❤❥❪❬❲♦ê❥ï➳❲♦❲♦❣❳❘❥❨❭❲♦❤ ò ❣❳❘ ②✫❪❬ê❥❣❳❲❃❨❬❲ù❩❭❪❹❲✰❛➷÷❥❪❫❛q❲ ❻✟õ ❪❫♠➭î❳☎❲♣t✕ê❥❲✕❦❥❩❭❻♦❲♦❣❳❩✛❘❚í➐❩❬❣❜❙♥❤❚❨æ❲♦❪❬❨❬ê❉❩æ❛q❛ ❪
í➐❦❥❘❚❻✟❪❫❙❱❣❳❘ g(x) = xk(x) ð ê❥❲♦❨❭❲ k ❙❱❩✣❻♦❣❜❘❿❪❬❙♥❘❿❦❥õ ❣❳❦❚❩❬♠❱î✕❤❚❙❱ë▲❲♦❨❬❲✟❘⑨❪❫❙↔❛qö❥♠♥❲❇❛❜❘❥❤ k(x) >
②✛❣❳❨❭❲♦❣➠❯❳❲✟❨♦♣
0, x 6= 0 õ
d
x k(x) > 0, x 6= 0
dx
❣❳❨
x

d
k(x) < 0,
dx

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✟❪❬❙♥❣❳❘
í➐❣❳❨❷❪❫ê❚❲✯❩❭î➅❩❭❪❬❲♦ò x + g(x) = 0 õ ê❥❲♦❘ g ❙♥❩✖❙♥❘✕❛q❤❥❤❥❙❱❪❬❙♥❣❳❘✒❤❥❙❱ë▲❲♦❨❭❲♦❘⑨❪❫❙♥❛❜ö❥♠♥❲■❪❫ê❥❲✟❩❬❲
T
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

û➈ü⑨ý❷þóÿ❷û➯♣✁❜⑩❳⑩✂➽❡t❣ õ ❥⑧ ♣❥ï õ

❛❜❘❚❤✫❤❚❲✔✓❉❘❥❲
F (s) =

Z

∞

u′ (t + s)f (t)dt.

s ♠♥❩❬❣❚♣ ✂ r ☎✬❣❳ö❥❩❭❲♦❨❭❯❜❲♦❩❵❪❫ê❉❛➷❪✖❙❱íPí➐❣❜❨❷❩❬❣❳ò✙0❲ s , F (s ) = 0 ð ê❥❙♥♠❱❲ F ′(s ) 6= 0 ❪❫ê❥❲✟❘
☎
í➐❣❳❨✇❩❬❦ ✁ ❻♦❙❱❲♦❘⑨❪❫♠➭îù❩❬ò❇❛❜♠❱♠ ǫ > 0 ❪❬ê❥❲♦❨❭❲✯❲❸0÷ø❙♥❩❞❪❫❩■❛❜❘ 0 ω ❝ôï▲❲✟❨❬❙❱❣ø❤❥❙❱❻ ❩❭❣❳♠❱❦❚❪❫❙❱❣❳0❘ v(t, ǫ) ❣❜í
❪❬ê❥❲❵ï▲❲✟❨❭❪❫❦❚❨❬ö➳❲♦❤✫❲♦ì❿❦❉❛q❪❬❙♥❣❳❘
x′′ + g(x) = ǫf (t) = ǫf (t + ω)

✂

✄✆☎❐è✞✝✩á✠✟óàæ✡ç ✟☞☛ àæä à✡ã à✍✌✎✟✡☎❐è✏✝✮á✠✟óàæç✡✟✦ã✒✑✔✓✔✟✲â✝è✟ã✙ä✡è♦ç✕✝✩ã✒✖♦å✍✌
á●è✟ã à✗✝ ã✒✑
Eǫ
✚✘ ✙✜✛ ✢ ✣✥✤✦✣★✧✪✩✬✫✮✭✰✯✱✧✲✣✥✳✴✧✶✵✎✧✲✭✸✷✥✹✺✯

s➯❻✟❻♦❣❳❨❭❤❥❙♥❘❥ú■❪❫❣✣❴ õ ✇❛❜❨❞❪❫ò❇❛❜☛
❘ ✡ ✂ ☎ ♣➅ï õ ⑥❳⑦✭✌❸♣❿❲✟ì➅❦❥❛q❪❫❙❱❣❳❘ ✡ 1 ✌✗❙♥❘✙ú❳❲✟❘❥❲♦❨❬❛❜♠▲❤❚❣ø❲✟❩➈❘❥❣q❪éê❉❛✌❯❜❲
❛✯❘❥❣❳❘✫❻♦❣❜❘❥❩❭❪❫❛❜❘⑨❪☎ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻✇❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘✮♣❚❲❸❯❳❲✟❘✒❙❱í xg(t, x, xǫ′) > 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 + x3 + ǫf (t, x, y),

❩❬❛q❪❫❙❱❩❭í î➅❙♥❘❥ú➈❪❫ê❥❲✗í➐❣❳♠❱♠♥❣ ð ❙♥❘❥ú➈❻♦❣❳❘❚❤❥❙❱❪❬❙♥❣❳❘❥❩➳í➐❣❳❨
ð ❙➭❪❫ê
f (0, 0, 0) = 0,

❦❥❘❚❙❱í➐❣❳❨❭ò❇♠➭î❇❙❱❘

ǫ>0

φ ∈ C 1 (R3 ),

f (t+1, x, y) = f (t, x, y),

f (t, x, y) = 0 if xy = 0

φ
→ ∞ when
x

x→∞

(t, y) ∈ R2 ,
δf
> 0 if xy > 0,
δy

and

δf
= 0 otherwise.
δy

❜❯ ❲♦❨❭❙❱í î➅❙♥❘❥ú | x |< ǫ, | y |< ǫ.
✐❢❘✣í❖❛q❻✟❪✌♣ ð ❲éê❉❛✌❯❳❲ xf (t, x, y) ❛❜❘❥❤ yf (t, x, y)) > 0 ❙➭í
❛❜❘❚❤ φ = 0 ❣q❪❫ê❥❲✟❨ ð ❙♥❩❭❲ õ
x, y

xy > 0, | x |< ǫ, y

❛❜❨❬ö❥❙➭❪❫❨❬❛❜❨❭î

û➈ü⑨ý❷þóÿ❷û➯♣✁❜⑩❳⑩✂➽❡t❣ õ ❥⑧ ♣❥ï õ ⑥

þ➯ê❥❲tí➐❦❥❘❥❻✟❪❬❙♥❣❳❘ V = 2x2 + x4 + 2x′2 ❩❫❛q❪❬❙♥❩ ✓❥❲♦❩ V ′ = −4ǫx′ f (t, x, x′) ♣➅❩❬❣ ❪❬ê❉❛q❪
❙❱í

❛❜❘❥❤ ′
❣❜❪❬ê❥❲♦❨ ð ❙♥❩❬❲
V ′ < 0 xx′ > 0, | x |< ǫ
þ➯ê❿❦❥❩ x ❻✌❛❜❘❚❘❥❣❜❪tö▲❲❵ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻❷❦❚❘❥V♠♥❲✟❩❬❩ = V0 ′ = 0 õ õ

➯þ 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❪❫❙❱❣❳❘ E õ þ✝❣
ǫ
❪❬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

✫✮✭ ✫✮✣☛✡✥✧✂☞ ✧✲✣☛✡ ✧✲✣ ✯ ✤✍✌

✉é❣❳❘❚❩❬❙♥❤❚❲♦❨t❛❜❘æ❲♦ì❿❦❉❛q❪❬❙♥❣❳❘æ❣❜í✵❪❫ê❚❲❵❪❢î➅ï➳❲

x′
✡ ✌

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.

þ➯ê❉❛q❪tò✙❲✌❛❜❘❚❩✲í➐❣❳❨

ǫ=0

⑥

✡ ✭✌

❪❫ê❥❲■❛❜❦❚❪❬❣❳❘❥❣❳ò✙❣❳❦❥❩➯❩❞î➅❩❭❪❫❲✟ò


x′ = 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❩❬❛✌î c ♣❀❛❜❘❚❤
þ➯ê❚❲✣ï➳❲♦❨❬❙❱❣ø❤✡í➐❦❚❘❥❻✟❪❬❙♥❣❳❘ T (c) ❤❥❲♦ï➳❲♦❘❚❤❥❙♥❘❥ú❇❣❳❘ c ❙❱❩ó❪❬ê❥❲■ò❇❙❱❘❥❙♥ò❇❛❜♠➳ï▲❲✟❨❬❙♥❣➅❤
γ ≡ γ(c).
❣❜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
û➈ü⑨ý❷þóÿ❷û➯♣✁❜⑩❳⑩✂➽❡t❣ õ ❥⑧ ♣❥ï õ ⑧

☎◗ ❩❭❙♥❘❥ú ❛✖❯❳❲♦❨❭❩❬❙❱❣❳❘❐❣❜í▲❪❫ê❥✆❲ ✓❚÷ø❲♦❤❇ï▲❣❜❙♥❘⑨❪✑❪❫ê❥❲✟❣❳❨❬❲✟ò ❤❥❦❥❲ó❪❫❣ ✎ õ ÿ☎❙♥❘❥ú❥♣❿❴ õ ✠é❦❚❪❬❪❫❛✗✑✮✑✟❣❳❘❥❙
❛❜❘❚❤✛s ❺❉❣❳❘❚❤❉❛ ✂ ✠❅❝❢❺ ☎●ï❥❨❬❣➠❯❜❲♦❤✒❪❬ê❉❛q❪ó❪❫ê❥❲✟❨❬❲ ❛❜❨❬❲■ï➳❲♦❨❬❙❱❣ø❤❚❙♥❻✖❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘❥❩➯❣qí✆✡ ✗✌➯ï❚❨❬❣➠❯➅❙♥❤❥❲✟❤
❪❬ê❉❛q❪➯❪❫ê❚õ ❲ ï➳❲♦❨❭❙♥❣➅❤❀í➐❦❥❘❥❻✟❪❬❙♥❣❳❘ T = T (c) ❣qí✝❪❬ê❥❲✣❛q❦❚❪❫❣❳❘❚❣❳ò❇❣❜❦❥❩ó❛❜❩❬❩❭❣ø❻✟❙↔❛q❪❬❲♦❤æ❩❭î➅❩❞❪❫❲♦ò ❙❱❩
ò✙❣❳❘❥❣❜❪❬❣❳❘❥❲t❛❜❘❥❤✙❪❫ê❉❛➷❪ ǫ ❙♥❩✑❩❬ò❇❛❜♠♥♠❉❲✟❘❥❣❳❦❥ú❜ê õ 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) õ
①ø❣❚♣✎❛❜❘❚❣❜❪❫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❥❣ ❛❜❘✕❛q❘❉❛❜♠♥❣❜ú❳❣❳❦❥❩☎❨❬❲♦❩❭❦❥♠❱❪✇❪❫❣❀❪❫ê❚❲✯ï❥❨❬❲✟❻♦❲♦❤❚❙♥❘❥úæ❣❳❘❥❲✯❦❚❘❥❤❥❲♦❨
ò✙❣❳❨❬❲❇❨❬❲✟❩❭❪❫❨❭❙♥❻❸❪❫❙❱❯❜❲❐ê⑨î➅ï▲❣q❪❫ê❥❲✟❩❬❲♦ð ❩ 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
′

✎

′

éç ã 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 ✣☎ þ➯ê❥❲✯ï➳❲♦❨❞❪❫❦❥❨❭ö❉❛q❪❬❙♥❣❳❘ù❙❱❩❷❩❬❦❥ï❚ï▲❣❳❩❭❲♦❤✕❪❫❣✡ö➳❲☛✡ ❻✟❣❳❘ø❝
❪❬❨❬❣❳♠❱♠↔❛❜ö❥♠➭î ï▲❲✟❨❬❙♥❣➅❤❥❙❱❻ ➭♣t❙ õ ❲ õ ♣☎❙❱❪✡❙❱❩✡ï▲❲✟❨❬õ ❙❱❣ø❤❥❙❱❻ ð ❙❱❪❬ê ❛ ï➳❲♦❨❭❙♥❣➅❤ ð ê❥❙♥❻❹ê ❻✌❛❜❘ ö➳❲✕❻❹ê❥❣❳❩❬❲✟❘
❛❜ï❚ï❥❨❬❣❳ï❚❨❬❙↔❛➷❪❫❲♦♠➭î ◗☎❘❥❤❚❲♦❨❑❯❳❲✟❨❭î✺ò✙❙♥♠❱❤ñ❻♦❣❜❘❥❤❥❙❱❪❬❙♥❣❳❘❚❩❑❙❱❪✯❙❱❩❑ï❥❨❭❣✻❯❜❲♦❤ ❪❫ê❉❛➷❪✯❪❫❣ù❲✌❛q❻❹ê ❩❭ò❐❛❜♠❱♠
❲✟❘❥❣❳❦❥ú❳ê✛❛❜ò✙ï❥♠♥❙➭õ❪❫❦❥❤❚❲✇❣❜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✡♠♥❣❜❪❫ú❳ê❥❙♥❙❱❻♦❩➯❛❜ï▲♠♥♠➭❲✟î☛❨❬❙❱ò❇❣ø❤ ❛➠õî❃❲✰÷❚ï➳❲♦❻❸❪✯❪❫ê❉❛➷❪✣❪❫ê❚❲✡❺➋❛❜❨✸✖➷❛❜❩ ò❇❲❸❪❫ê❥❣➅❤ñ❻✌❛q❘✺ö▲❲✡❛qú⑨❛❜❙♥❘➩❛❜ï❥ï❥♠❱❙♥❲✟❤ í➐❣❳❨
✎ï➳❲♦❨❞❪❫❦❥❨❭ö▲❲✟❤✺r✩❙♥❲♦❘❥❛❜❨❬❤ñ❲✟ì❿❦❉❛q❪❫❙❱❣❳❘❥❩ þ➯ê❥❙❱❩✣ê❉❛❜❩ ö▲❲✟❲♦❘✺❻✟❣❳❘❥❩❬❙❱❤❥❲♦❨❭❲♦❤ ❛❜❘❚❤ñï❥❨❬❣➠❯❜❲♦❤✺❙♥❘❃❪❫ê❥❲
❛❜❦ø❪❫❣❳❘❥❣❜ò❇❣❳❦❚❩➯❻✌❛❜❩❭❲■ö⑨îæ❺➋❛❜❨ ✖➷❛❜❩➯ê❥õ ❙♥ò✙❩❬❲✟♠❱í ✂ ❺ ✄ ☎ õ
✞ ❦❥❨✝ï❥❨❬❣➅❣❜í ð ❲✲ú❳❙❱❯❜❲éê❥❲✟❨❬❲➯ò❇❛❜❤❥❲➈❙❱❪✝ò❇❣❳❨❭❲é❩❭❙♥ò✙ï❥♠♥❲❅❛❜❘❥❤✯❻✟❣❳❘⑨❪❹❛❜❙❱❘❥❩✬❩❭❣❳ò✙❲✲❙♥ò✙ï❥❨❬❣➠❯❜❲♦ò✙❲♦❘⑨❪❫❩
û➈ü⑨ý❷þóÿ❷û➯✁♣ ❜⑩❳✂⑩ ➽❡t❣ õ ⑧❥♣❥ï õ ⑤
✁

❱❙ ❘☛❦❥❩❬❙❱❘❥ú❀❙❱❘☛ï❉❛❜❨❞❪❫❙♥❻✟❦❥♠↔❛q❨☎ò✙❲✟❪❬ê❥❣ø❤❚❩✇❣❜í ✂ ❺ ✄ ☎✵❪❫❣✡❲✟❩❭❪❫❙❱ò❐❛➷❪❫❲■❲❸÷ø❙♥❩❞❪❫❲♦❘❚❻♦❲➽❨❬❲✟ú❳❙♥❣❳❘❚❩✇❣❜í✑ï➳❲♦❨❭❙➭❝
❣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❦❉❛➷❪❫❙♥❣❜❘
u′′ + f (u)u′ + g(u) = 0.

(L)

✐❢❘æ❣❳❨❭❤❥❲♦❨➯❪❫❣➽ê❉❛✌❯❜❲ ❛✯❦❥❘❚❙♥ì❿❦❥❲❵ï▲❲✟❨❬❙❱❣ø❤❥❙❱❻❷❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘ ð ❲■❩❬❦❥ï❥ï➳❣❳❩❭❲❵❪❬ê❥❲✖í➐❦❥❘❥❻❸❪❫❙♥❣❜❘❥❩
❛❜❨❬❲✖❣❜í✝❻♦♠♥❛❜❩❬❩ C 2 õ þ➯ê❥❲ ❙❱❘⑨❪❫❲♦ú❜❨❫❛❜♠❱❩
g
F (x) =

❣❜í

❛❜❘❚❤

Z

x

0

f (t)dt,

G(x) =

Z

0

f

❛❜❘❚❤

x

g(t)dt

❨❬❲✟❩❬ï➳❲♦❻❸❪❫❙❱❯❜❲♦♠➭î■❛❜❨❬❲✗❩❭❦❥❻❹ê ❪❬ê❉❛q❪ lim F (x) = ∞, ❛❜❘❥❤ lim G(x) =
x→∞
x→∞
❉
ê
❜
❛
■
❩
✡
❛
❥
❦
❥
❘
♥
❙
❿
ì
❥
❦
✞
❲
✑
✟
♦
❲
❬
❨
❣
➯
þ
❚
ê
♦
❲
❃
❘
❱
❙
❵
❪
♥
❙
❩
❿
✖
❥
❘
❣
☞
❘
✂ ✉❅❝❢✻r ☎✗❪❫ê❉❛q❪
ð
✡❖r ✌éê❥❛❜❩t❛➽❩❞❪❹❛❜ö❚♠♥❲❵❘❥❣❳❘æ❻♦❣❜❘❥F❩❭❪❫❛❜❘⑨❪✇ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻☎❩❬❣❳♠❱❦❚❪❫❙❱❣❳õ❘ u (t) ð ❙❱❪❬ê✛ï➳❲♦❨❭❙♥❣➅❤ τ õ
û✗ì❿❦❉❛q❪❬❙♥❣❳❘ ✡ r ✌✬❙❱❩✗❦❥❩❬❦❉❛q♠♥♠❱î ❩❭❪❬❦❥❤❥❙❱❲♦❤✙ö❿î❑ò❇❲♦❛❜❘❥❩✗❣❜í✎❛❜❘➽❲♦ì❿❦❥0❙➭❯❜❛q♠♥❲♦❘⑨❪✑ï❥♠♥❛❜❘❥❲➯❩❞îø❩❞❪❫❲✟0ò õ þ➯ê❥❲
ò✙❣❳❩❭❪ó❦❚❩❬❲♦❤✫❣❳❘❥❲✟❩✇❛q❨❬❲ ✎

f
g
ô
✐
■
❪
❱
❙
❵
❩
❜
❛
❩❬❩❭❦❥ò❇❲✟❤❃❪❬ê❉❛q❪
∞.

❛❜❘❥❤✛❛❜♠❱❩❬❣



✡⑤ ✌

u′ = v
v ′ = −g(u) − f (u)v



u′ = v − F (u)
v ′ = −g(u)

✡✞✝✭✌

✐❢❘æí❖❛❜❻✟❪♦♣❚❪❫ê❥❲❸î✫❛❜❨❬❲✖❲✟ì❿❦❥❙❱❯q❛❜♠❱❲♦❘⑨❪ó❪❫❣✯❪❫ê❥❲ ✻❝ô❤❥❙♥ò✙❲♦❘❚❩❬❙♥❣❜❘❉❛❜♠➳❩❭î➅❩❭❪❬❲♦ò
(S)

❛qí ❪❬❲♦❨➯❙❱❘❿❪❬❨❬❣➅❤❥❦❥❻✟❙♥❘❥ú❑❪❬ê❥❲❵❘❥❣❜❪❹❛➷❪❫❙♥❣❜❘❥❩


x˙ = h(x)

x = col[x1 , x2 ]

x1 = −u(t)
˙ − F (u(t))
x2 = u(t)

✡③ ✌
û➈ü⑨ý❷þóÿ❷û➯♣✁❜⑩❳⑩✂➽❡t❣ õ ❥⑧ ♣❥ï õ ✝

ð ê❥❲✟❨❬❲
①ø❦❚ï❥ï▲❣❜❩❬❲

x = col[x1 , x2 ]

❛❜❘❥❤

h(x) = col[g(x2 ), −x1 − F (x2 (t)).

u0(0) = a,

❩❭❣✙❪❬ê❉❛q❪➯❪❫ê❚❲■ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻✇❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘✡❣❜í✝ï▲❲✟❨❬❙♥❣➅❤

u′0 (0) = 0 > 0
τ0

❣❜í✝❪❫ê❥❲❷❯q❛❜❨❭❙↔❛q❪❬❙♥❣❳❘❥❛❜♠▲❩❞î➅❩❭❪❫❲✟ò

y˙ = h′ x (p(t))y

❙❱❩
p(t)
˙ = col[g(u0(t)), u˙ 0 (t)],

ð ê❥❲✟❨❬❲

p(t) = col[−u˙ 0 (t) − F (u0 (t)), u0 (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
t
u¨ + f (u)u˙ + g(u) = ǫγ( , u, u)
˙
τ

(LR )

ð ❥ê ❲✟❨❬❲ t ∈ R, ǫ ∈ R ❙❱❩➯❛✯❩❬ò❇❛❜♠♥♠▲ï❉❛❜❨❬❛❜ò❇❲❸❪❫❲✟❨♦♣ | ǫ |< ǫ0 ,
❩❭❦❥❻❹ê✛❪❬ê❉❛q❪
í➐❣❜❨➯❩❬❣❳ò✙❲
| τ − τ0 |< τ1
0 < τ1 < τ2 õ
②✛❣❜❨❬❲♦❣➠❯❜❲♦❨♦♣❉❪❬ê❥❲❵❻♦♠♥❣❜❩❬❲♦❤✫❣❳❨❭ö❥❙❱❪

τ

❙♥❩ó❛❑❨❭❲✌❛❜♠✮ï❥❛❜❨❫❛❜ò✙❲✟❪❬❲♦❨

0

{(u, v) ∈ R2 : u(t) = u0 (t), v(t) = u˙ 0 (t), t ∈ [0, τ0 ]}

ö➳❲♦♠❱❣❳❘❥ú❳❩✲❪❬❣➽❪❬ê❥❲❵❨❬❲✟ú❳❙♥❣❳❘
2
u2 + v 2 < r 2 }.
✐❢❘æ❪❫ê❥❲❵❩❬❛❜ò❇❲ ð ❛✌î✫❛❜❩✲í➐❣❳❨ {(u, ♣❥v)❪❫ê❚∈❲ R➠❝❢❤❥:❙❱ò❇
❲✟❘❥❩❬❙❱❣❳❘❉❛❜♠▲❲♦ì❿❦❥❙❱❯q❛❜♠❱❲♦❘⑨❪ó❩❭î➅❩❞❪❫❲♦ò✦í➐❣❜❨
(L)

(SL )

ð ê❥❲✟❨❬❲

(LR )

❙♥❩

t
x˙ = h(x) + ǫq( , x)
τ

q = col[q1 , q2 ],


q1 = −γ( τt , x2 , −x1 − F (x2 ))
q2 = 0

✡✁✭✌

û➈ü⑨ý❷þóÿ❷û➯♣✁❜⑩❳⑩✂➽❡t❣ ❥⑧ ♣❥ï ③
õ
õ

✩✬✫✮✭ ✯✱✧✲✣✥✳✪✧ ✤✍✌

✬✙ ✘

☞ ✧ ✵✎✫✮✤ ✡ ✫✜✳ ✭✱✤✦✹✮✷ ✯✱✫✮✤✦✣✥✭ ✤✍✌

(LR )
❡ ❣ ð ð ❲ ð ❙♥♠♥♠✝❦❥❩❬❲❑❪❬ê❥❲➽❴✝❣❜❙♥❘❥❻♦❛❜✠❨ ✟❑ò✙❲✟❪❬ê❥❣➅❤☛í➐❣❳❨✇❪❫ê❚❲✯❤❥❲✟❪❬❲♦❨❬ò✙❙♥❘❥❛q❪❫❙❱❣❳❘☛❣❜í✑❪❫ê❚❲➽❛❜ï❥ï❥❨❭❣➠÷❿❝
t
❙❱ò❐❛q❪❬❲ ❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘☛❣❜í✗❪❫ê❚❲❑ï▲❲✟❨❭❪❬❦❥❨❬ö➳❲♦❤✕❲♦ì❿❦❉❛➷❪❫❙♥❣❜❘
þ➯ê❥❲✯❲✰÷❚❙❱❩❭❪❬❲♦❘❥❻✟❲✯❣❜í✑❪❫ê❥❲✣í➐❦❚❘❥❤❉❛➷❝
(LR ) õ
ò✙❲♦❘⑨❪❹❛q♠✵ò❇❛q❪❬❨❬❙➭÷✡❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘✛❣qíP❪❫ê❥❲✘✓❉❨❬❩❞❪✇❯q❛❜❨❭❙↔❛q❪❬❙♥❣❳❘❉❛q♠✮❩❭î➅❩❭❪❬❲♦ò✷❣❜í
❛❜❘❚❤✛❪❫ê❚❲
❦❥❘❚❙♥ì❿❦❥❲❵ï▲❲✟❨❬❙❱❣ø❤❥❙❱❻❷❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘
❻✟❣❳❨❬❨❭❲♦❩❭ï▲❣❳❘❚❤❥❙♥❘❥ú✯❪❬❣
❙❱❩ó❛❜❩❬x˙❩❭❦❥=ò❇h(x)
❲✟❤
✐❢❘❃❣❳❨❬❤❚❲♦❨✖❪❫❣✫ú❳❲✟❪■❲✟❩❭❪❫❙❱ò❐❛➷❪❫❲♦❩❷p(t)
í➐❣❜❨❵❪❬ê❥❲➽❲❸÷ø❙♥❩❞❪❫❲✟❘❥❻♦❲❇❣❜íéï➳❲♦u❨❭0❙♥❣➅(t)
❤❥❙♥❻✯❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘❥❩ õ ð ❲✙ê❉❛✌❯❳❲➽❪❫❣
❻♦❛❜♠♥❻✟❦❥♠↔❛q❪❬❲➽❩❬❣❳ò✙❲❇❻✟❣❳❘❥❩❞❪❹❛❜❘⑨❪❫❩ ❺➋❣❜♠♥♠♥❣ ð ❙❱❘❥úæ❺➋❛❜✸❨ ➷✖ ❛❜✄
❩ ✂ ❺ ☎ ♣✝❪❫ê❚❲❀ü❳❛❜❻✟❣❳ö❥❙❅ò❐❛➷❪❫❨❬❙➓÷
ê❉❛❜❩
J
õ
✄
❪❬ê❥❲✖í➐❣❳♠♥♠❱❣ ð ❙❱❘❥ú✣í➐❣❳❨❭ò
✂

J(τ0 ) = −I +



g(a) 0
0
0



+ Y (τ0 )

❜❛ ❘❥❤
❙♥❩➽❪❫ê❥❲✛í➐❦❥❘❥❤❉❛❜ò✙❲♦❘⑨❪❫❛❜♠ó❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘ ò❇❛q❪❬❨❬❙➭÷✺❣❜í❷❪❫ê❚❲✛❯q❛❜❨❫❛➷❪❫❙♥❣❜❘❉❛❜♠
Y (t)
❩❞îø❩❞❪❫❲✟ò ð ❙❱❪❫ê
Y (0) = I

I = Id2

y˙ =



O
g ′(u0 (t))
y.
−1 −f (u0 (t))


✡ ⑦✭✌

ô✐ ❪■❙♥❩✖ï❥❨❭❣➠❯❳❲♦❤✎♣✵❪❬ê❉❛q❪❵❙❱í
❪❫ê❥❲✟❘❃❪❬ê❥❲♦❨❭❲✙❲✰÷ø❙♥❩❭❪❵❦❚❘❥❙♥ì❿❦❥❲✟♠❱îù❤❥❲❸❪❫❲✟❨❬ò✙❙♥❘❥❲✟❤
detJ(τ0 ) 6= 0
í➐❦❥❘❚❻✟❪❫❙❱❣❳❘❥❩
❛❜❘❚❤
❤❥❲✔✓❥❘❥❲♦❤❇❙♥❘✯❪❫ê❥❲ó❘❥❲✟❙♥ú❳ê❿ö➳❣❳❨❬ê❚❣ø❣➅❤➽❣❜í
❩❭❦❥❻❹ê❇❪❬ê❉❛q❪
h(ǫ, φ)
(0, 0)
❪❬ê❥❲✖í➐❦❥❘❥❻✟❪❬❙♥❣❳τ❘ (ǫ, φ)
u(t; φ, p0 + h(ǫ, φ), ǫ, τ (ǫ, φ))

❙❱❩❑❛✒ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻➽❩❬❣❜♠♥❦❚❪❬❙♥❣❳❘ ❣❜íó❩❭î➅❩❭❪❬❲♦ò
❙❱❪❬ê
❛❜❘❥❤
(SL ) ð
τ (0, 0) = τ0
②✛❣❜❨❬❲♦❣➠❯❜❲♦❨♦♣P❛❜❘ ❲✟❩❭❪❬❙♥ò❇❛q❪❫❲➽❙♥❩■ú❳❙➭❯❳❲♦❘❃í➐❣❳❨✖❪❫ê❥❲✙❨❬❲✟ú❳❙♥❣❜❘ ❙❱❘ ð ê❥❙❱❻❹ê ❪❬ê❥❲❇❪❬ê❥❲✙h(0,
❯q❛❜❨❭❙↔❛❜0)ö❚♠♥❲♦=❩ 0 õ
❛❜❘❚❤
ò❇❛✌î✛❯q❛❜❨❞î ❺❉❣❳❨✇❲❸÷❚❛❜ò✙ï❥♠♥❲❑❙❱❘ù❲✟❯q❛❜♠❱❦❉❛q❪❫❙❱❘❥ú❐❪❬ê❥❲✯❘❥❣❳❨❭ò✸❣❜í❅❪❫ê❥❲✯❤❚❙❱ë▲❲♦❨❬❲✟❘❥❻♦❲✯❣qǫí
ü❳❛❜❻✟❣❳ö❥φ❙✩ò❇❛q❪❫❨❭❙♥❻✟❲♦❩ õ
J(ǫ, φ, τ, h) − J(0, 0, τ0 , 0).

s ❪❬❨❬❙❱❯➅❙♥❛❜♠ ❝ôï▲❲✟❨❬❙♥❣➅❤❥❙❱❻t❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘❐❣❜í ✡❖⑦✭✌➈❙♥❩
s ❪❬❨❬❙➭❯ø❙♥❛❜♠➋❻✌❛❜♠❱❻♦❦ø❝
τ0
col[g(u0(t)), u˙ 0 (t)].
♠♥❛q❪❫❙❱❣❳❘❀ú❳❙➭❯❳❲♦❩➯❪❬ê❥❲❵❣❜❪❫ê❥❲✟❨ó♠♥❙♥❘❚❲✌❛❜❨❭♠❱î✙❙♥❘❥❤❚❲♦ï➳❲♦❘❥❤❥❲✟❘⑨❪✇❩❭❣❳♠♥❦❚❪❬❙♥❣❳❘✡❣qí ✡❖⑦✭✌
col[g(u0 (t))v(t), u0(t)v(t)
˙
+

ð ê❥❲✟❨❬❲
í➐❣❳❨

v(t) =

Z

0

t

−2 ′

[g0 (s)] g (u0 (t))exp[−

g(u0(t))
v(t)]
˙
g ′(u0 (t))
Z

s
0

f (u0 (σ))dσ]ds

t ∈ [0, τ0 ] õ

û➈ü⑨ý❷þóÿ❷û➯♣✁❜⑩❳✂⑩ ➽❡t❣ ❥⑧ ♣❥ï
õ
õ

þ➯ê❥❲✟❘✛❪❫ê❚❲❵í➐❦❚❘❥❤❉❛❜ò✙❲♦❘⑨❪❹❛q♠✮❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘✡ò❇❛q❪❫❨❭❙➭÷❐❣❜í ✡❖⑦✭✌ ð ❙❱❪❬ê


Y (t) = 

g(u0 (t))
g(a)
u˙ 0 (t)
g(a)

s➯❻✟❻♦❣❳❨❭❤❥❙♥❘❥ú ❫❪ ❣ r✩❙♥❣❜❦❚❯➅❙♥♠♥♠❱❲
❙❱❩➯ú❳❙❱❯❜❲♦❘æö⑨î
W (o) = 1

þ➯ê❥❲❵❻❹ê❉❛q❨❫❛❜❻❸❪❫❲♦❨❭❙♥❩❞❪❫❙♥❻❵ò❑❦❚♠❱❪❫❙❱ï❥♠♥❙❱❲♦❨❭❩➈❣❜í ✡ ⑦ ✲✌ ❛❜❨❬❲
❙➭í✝❛❜❘❥❤✫❣❳❘❥♠❱î❐❙❱í

Z

0

Y (τ0 ) =

þ➯ê❿❦❥❩✟♣ ð ■
❲ ú❳❲✟❪

þ➯ê❥❲✟❨❬❲❸í➐❣❳❨❬❲q♣

J

=



ρ1 = 1
Z

❛q❘❥❤

τ0
0

f (u0 (τ ))dτ ].

✡ ✭⑩ ✌

✄

f (u0 (τ ))dτ > 0.





ð ❙❱❪❬ê

f (u0 (τ ))dτ ].

τ0

þ➯ê❥❲■❙♥❘❥❙➭❪❫❙♥❛❜♠➋❻♦❣❳❘❥❤❚❙❱❪❫❙❱❣❳❘❥❩➯ú❳❙➭❯❳❲ 0

J=

W (t)

t

ρ2 = W (τ0 ) = exp[−

−1

.

0 (t))
g(a)u˙ 0 (t)v(t) + g(a) gg(u
˙
′ (u (t)) v(t)
0
❩✡í➐❣❳❨❬ò❑❦❚♠↔❛ ❪❬ê❥❲ ✎ ❨❬❣❜❘❥✸❩ ❿✖ ❙↔❛q❘④❤❥❲✟❪❬❲♦❨❬ò✙❙♥❘❥❛❜❘⑨❪

Z

❙♥❩



g(a)g(u0(t))v(t)

W (t) = exp[−

ρ2 < 1

Y (0) = I

1 g 2 (a)v(τ0 )
.
0
ρ2


g(a) g 2 (a)v(τ0 )
,
0
ρ2 − 1


g −1(a) g 2 (a)v(τ0 )(1 − ρ2 )−1
.
0
−(1 − ρ2 )−1


|| J −1 ||= 2 max [g −1 (a), (1 − ρ2 )−1 , g 2(a)v(τ0 )(1 − ρ2 )−1 ].

þ➯ê❥❲■❙♥❘⑨❯❳❲✟❨❬❩❭❲ ò❇❛q❪❫❨❭❙➭÷❇❣❜í


Y −1 (t) = W (t) 

Y (t)

❙♥❩

0 (t))
˙
−g(a)g(u0(t))v(t)
g(a)u˙ 0(t)v(t) + g(a) gg(u
′ (u (t)) v(t)
0
0 (t)
− u˙g(a)

☎❡ ❣ ð ð ❲✖ê❉❛✌❯❳❲❷❪❫❣✣❤❥❲✟❪❬❲♦❨❬ò✙❙♥❘❚❲☎❪❫ê❚❲❷❻♦❣❳❘❚❩❭❪❹❛q❘❿❪❬❩✲í➐❣❳❨é❩❞î➅❩❭❪❫❲✟ò
♠❱❲✟❪ó❦❥❩ó❤❥❲✟❘❥❣❜❪❫❲

g(u0 (t))
g(a)

(LR ) õ



.

❺❉❣❳♠❱♠♥❣ ð ♥❙ ❘❥ú ✂ ❺

✄

S = {x = (x1 , x2 ) ∈ R2 / x2 2 + [−x1 − F (x2 (t))]2 < r 2 }

û➈ü⑨ý❷þóÿ❷û➯♣✁❜⑩❳⑩✂➽❡t❣ õ ❥⑧ ♣❥ï õ ⑦

☎



 g0

:= maxx∈S | g(x2 ) |,
g1 := maxx∈S | g ′ (x2 ) |,


g2 = maxx∈S | g ′′ (x2 ) | .

✡

✌

✄✂✄

f2 := maxx∈S | f ′ (x2 ) |

f1 := maxx∈S | f (x2 ) |,


 q0

:= maxx∈S,s∈R | q(s, x) |,
q1 := maxx∈S,s∈R | qx′ (s, x) |,


q2 := maxx∈S,s∈R | qs′ (s, x) | .

✡

✄

✌

K−1 := maxt∈[ −τ0 ,τ0 ] | Y −1 (t) | .

K := maxt∈[ −τ0 ,τ0 ] | Y (t) |,

þ➯ê❿❦❥❩✟♣ ð ■
❲ ò❐❛✌î❀❤❚❲♦❤❥❦❥❻✟❲■❪❬ê❉❛q❪
2

2

P := maxt∈[ −τ0 ,τ0 ] | p(t)
˙ |≤
2

K
.
2

þ➯ê❥❲❵❙❱❘❥❙❱❪❬❙↔❛❜♠➳ï❥ê❉❛❜❩❭❲ ❛q❘❥❤✡❪❫ê❥❲❵ï➳❲♦❨❬❙❱❣ø❤ ê❉❛✌❯❳❲✖❪❬❣✙❯❜❲♦❨❬❙➭í î❀❪❫ê❚❲❷í➐❣❳♠♥♠❱❣ ð ❙❱❘❥ú❥♣ ð ê❥❙❱❻❹ê✛❻♦❛❜❘
ö➳❲❵❲✌❛❜❩❭❙♥♠❱î❇❣❳ö❚❪❫❛❜❙♥❘❚❲♦❤✡φí➐❨❬❣❳ò ❪❫ê❥❲■❛❜ö➳❣➠❯❳❲■❲♦τ❩❞❪❫❙♥ò❇❛q❪❬❲♦❩♦♣❚❩❬❲♦❲ ✂ ❺ ☎
✄

φ<

✐ôí✝❙♥❘✫❛❜❤❥❤❚❙❱❪❫❙❱❣❳❘ ð ❲■❩❭❦❥ï❥ï➳❣❳❩❬❲

ǫ

τ0
,
2

❛❜❘❥❤

h

| τ − τ0 |<

❛❜❨❭❲■❩❭❦❥❻❹ê✛❪❬ê❉❛q❪

τ0
.
2

3
3
g0 | ǫ | + | h |< σexp(− g1 τ0 )
2
2

✡➐ê❥❲♦❨❭❲
❙♥❩❑❪❬ê❥❲æ❤❥❙♥❩❞❪❹❛❜❘❚❻♦❲æö▲❲❸❪ ð ❲✟❲♦❘ ❪❬ê❥❲æï❉❛q❪❬ê ❣❜ít❪❫ê❚❲✫ï➳❲♦❨❭❙♥❣➅❤❥❙♥❻❐❩❬❣❳♠❱❦❚❪❫❙❱❣❳❘ ❛❜❘❥❤ ❪❫ê❥❲
ö➳❣❳❦❥❘❥❤❥σ❛❜❨❭î✡❣qí ✌➈❪❫ê❚❲♦❘✛❛➽❩❭❣❳♠♥❦ø❪❫❙♥❣❜❘✡❣❜í
❲❸÷ø❙♥❩❞❪❫❩
õ
S
(LR )
þ➯ê❥❲✟❘✮♣ ð ❲❵ê❉❛✌❯❜❲ ï❥❨❭❣➠❯❳❲♦❤✫❪❫ê❥❲✖í➐❣❳♠❱♠♥❣ ð ❙♥❘❚ú

✂✁☎✄✝✆✟✞✠✄☛✡ ☞
✌✎✍ 1 ✵✑✏ ✪✒✴✏ ✵✔✓ ✱ ★✒✳ ✙✖✕ ✪✗✦ ✪ ✙✴✤ ✳✴✦ ✵✑✏✴✤ ✵✶✙✗✓✙✘ ★✰✤✧✵ ✱ ★✰✵✶✳✴✦ ✛ ✍✛✚✢✜✤✣✥✚ ✤✎✕ ✪✗✤
✓ ✳✸✪✗✢✦✏ ✵✩✢ ✳★✧✩✘ ✪✗★✰✵✩✤✧✯ ✚✢✪☛✫✖✣ ✕✥✛✣★✫✷✠✏ ✣ ✤✎✕✥✳✴✢ ✤✎✕✥✳✴✦ ✳✚✪✗✦ ✳ ✤✭✬ ✛ ✍ ✘ ✢ ✙✴✤ ✵✶✛✣✦✢ ✏ τ, h : U → R ✪✗✢ ✷
✪ ✙✜✛✣✦✢ ✴✏ ✤ ✪✗✢✥✤
✏✤✘✍✙✖✕ ✤✎✕ ✪✗✤ ✤✔✥✕ ✳✲✱✥✳✴✦✴✵✶✛✮✷✗✵✶✙✮✏ ✛✣★✯✘ ✤ ✵✶✛✣✢ u(t, φ, a + h(ǫ, φ), ǫ, τ (ǫ))
τ
✛ ✍ ✳★✧✩✘ ✪✗✤✧✵✶✛✣✢ τ1 < 2
0

✳✱✰ ✵✑✴✏ ✤✔✏ ✍ ✣✛ ✦

(LR )
(ǫ, φ) ∈ U ✲

u¨ + f (u)u˙ + g(u) = ǫγ(

✪✗✢ ✷

t
, u, u)
˙
τ (ǫ)

| τ − τ0 |< τ1 , τ (0, 0) = τ0 , h(0, 0) = 0.

û❅ü⑨ý❷þóÿ❷û➯♣✁❜⑩❳⑩ ✯❡t❣ ❥⑧ ♣❥ï ⑩
õ
õ ✄

✬✙
✧✲✳✴✫ ✹ ✳ ✭✸✧✲✭
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
✂

✂

☞

☎

☎

(LRA )

u¨ + f (u)u˙ + g(u) = ǫγ

ð ê❥❲✟❨❬❲ó❪❫ê❥❲tï➳❲♦❨❭❪❬❦❥❨❬ö❥❛q❪❫❙❱❣❳❘❇❙❱❩✑❙♥❘❥❤❥❲✟ï▲❲✟❘❥❤❥❲✟❘❿❪➈❣❳❘✙❪❫ê❥❲ó❪❫❙❱ò❇❲✲❯q❛❜❨❭❙↔❛❜ö❥♠❱❲
þ➯ê❥❲❵❲✟ì➅❦❚❙❱❯q❛❜♠♥❲✟❘⑨❪☎ï❚♠↔❛❜❘❥❲✖❩❞î➅❩❭❪❫❲✟ò ❙♥❩➯❣❜í✵❪❫ê❚❲❵í➐❣❜❨❬ò

γ ≡ γ(u, u,
˙ ǫ, τ ) õ

t
x˙ = h(x) + ǫq( , x)
τ

ð ❥ê ❲✟❨❬