Đạo hàm phân thứ caputo và sự tồn tại nghiệm của phương trình vi phân phân thứ

TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI 2
KHOA TOÁN
*************

NGUYỄN NGỌC HUYỀN

ĐẠO HÀM PHÂN THỨ CAPUTO VÀ SỰ
TỒN TẠI NGHIỆM CỦA PHƯƠNG TRÌNH
VI PHÂN PHÂN THỨ
KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC
Chuyên ngành: Giải tích

HÀ NỘI – 2018


TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI 2
KHOA TOÁN
*************

NGUYỄN NGỌC HUYỀN

ĐẠO HÀM PHÂN THỨ CAPUTO VÀ SỰ
TỒN TẠI NGHIỆM CỦA PHƯƠNG TRÌNH
VI PHÂN PHÂN THỨ
KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC
Chuyên ngành: Giải tích

Người hướng dẫn khoa học
TS. HOÀNG THẾ TUẤN

HÀ NỘI – 2018

ứ t ồ t ữớ ồ
ữủ rt sỹ q t ú ù ừ qỵ ổ

ỷ ớ ỡ t sỹ tr s s ố ợ
ồ ỏ st tố t
ữủ ồ t tr sốt tớ q t



t t ữợ ỳ tự
ổ ụ ữ ở số õ t t qt
t tr ữớ trữợ
r õ t ỏ s sõt rt s
ữủ ỵ õ õ ừ ổ õ t ồ t
ữủ ụ ữ tr ỗ t tự
t
ố ũ ú qỵ ổ ỗ sự ọ
t ổ tr sỹ qỵ
t ỡ
ở t


ồ


▲❮■ ❈❆▼ ✣❖❆◆
❑❤â❛ ❧✉➟♥ ♥➔② ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤ s❛✉ q✉→ tr➻♥❤ ❤å❝ ❤ä✐✱ ♥❣❤✐➯♥ ❝ù✉

❝õ❛ ❜↔♥ t❤➙♥ ❡♠ ✈➔ sü ❤÷î♥❣ ❞➝♥ ♥❤✐➺t t➻♥❤ ❝õ❛

❚✉➜♥ ♣❤á♥❣ ①→❝ ①✉➜t t❤è♥❣ ❦➯ ❱✐➺♥ ❚♦→♥ ❤å❝✳

❚❙✳ ❍♦➔♥❣ ❚❤➳

❚r♦♥❣ ❜➔✐ ❦❤â❛ ❧✉➟♥ ❝õ❛ ♠➻♥❤ ❡♠ ❝â t❤❛♠ ❦❤↔♦ ♥ë✐ ❞✉♥❣✱ ❦➳t q✉↔
❝õ❛ ♠ët sè ❜➔✐ ❜→♦ ♥÷î❝ ♥❣♦➔✐ ✈➔ ♠ët sè t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❦❤→❝✳ ❊♠
①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❜➔✐ ❦❤â❛ ❧✉➟♥ ❝õ❛ ♠➻♥❤✱ ❦❤æ♥❣ s❛♦ ❝❤➨♣ ❜➜t ❦➻
❜➔✐ ❦❤â❛ ❧✉➟♥ ♥➔♦ ❦❤→❝✳ ❊♠ ①✐♥ ❝❤à✉ ❤♦➔♥ t♦➔♥ tr→❝❤ ♥❤✐➺♠ ✈î✐ ❧í✐
❝❛♠ ✤♦❛♥ ❝õ❛ ♠➻♥❤✳
❍➔ ◆ë✐✱ ♥❣➔② ✵✾ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽
❙✐♥❤ ✈✐➯♥

◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥


▼ö❝ ❧ö❝
▲í✐ ♠ð ✤➛✉

✶

✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❣✐↔✐ t➼❝❤ ♣❤➙♥ t❤ù

✹

✶✳✶

❚➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷

✣↕♦ ❤➔♠ ♣❤➙♥ t❤ù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✶✳✷✳✶

✣↕♦ ❤➔♠ ♣❤➙♥ t❤ù ♥❣✉②➯♥ t❤õ② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✶✳✷✳✷

✣↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

✷ ❑❤✐ ♥➔♦ ♠ët ❤➔♠ ❝â ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù❄

✶✹

✷✳✶

P❤→t ❜✐➸✉ ❦➳t q✉↔ ❝❤➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✷✳✷

▼ët sè ❤➔♠ ❝â ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✷✳✷✳✶
✷✳✷✳✷

❳➨t ❤➔♠ v ∈ H β [0, T ] , 0 < α < β ≤ 1, v (0) = 0. ✶✻
❳➨t ❤➔♠ tα , α > 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✷✳✸

✣↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✷✳✹

❈❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ❝❤➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

✷✳✹✳✶

❙ü ❦❤↔ ✈✐ ❝õ❛ J 1−α v0 . ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

✷✳✹✳✷

▼ët ✈➔✐ t➼♥❤ ❝❤➜t ❝õ❛ t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù J α u✳

✷✺

✷✳✹✳✸

❙ü t÷ì♥❣ ✤÷ì♥❣ ❝õ❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✐✮✱ ✭✐✐✮✱ ✭✐✐✬✮✱
✭✐✐✐✮✱ ✭✐✐✐✬✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✐✐

✸✷


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❑➳t ❧✉➟♥

✸✸

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✺

✐✐✐


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

✶


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❇❷◆● ❑➑ ❍■➏❯
❑➼ ❤✐➺✉ ❚➯♥ ❣å✐
R

❚➟♣ sè t❤ü❝

Rn

❑❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ t❤ü❝ n ❝❤✐➲✉

C

❚➟♣ ❤ñ♣ ❝→❝ sè ♣❤ù❝

|z|

●✐→ trà t✉②➺t ✤è✐✭♠♦❞✉❧❡✮❝õ❛ sè t❤ü❝ ✭♣❤ù❝✮ ③

·

❈❤✉➞♥ ❝õ❛ ♠ët ✈➨❝ tì ❤♦➦❝ ♠❛ tr➟♥

L1 [a, b]

❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b]

C m [a, b]

❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ ❝➜♣ m tr➯♥ ✤♦↕♥ [a, b]

H µ [0, T ] ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ ❍☎♦❧❞❡r
Γ

❍➔♠ ●❛♠♠❛

✶



ố t tớ t t tr ỡ s ừ
t t ợ t

dn
f (x)
dxn
ừ ởt số f tổ

tr ởt ự tữ ỷ st r ự tữ ỗ
dn
1
st ọ t s
f
(x)


n
=
ự tữ
dxn
2
ỗ ữủ t ừ st
ữủ tứ ữ sỹ t sỹ t ừ
tự rt rt ỳ
tự rt ữủ sỷ ử tố sỷ ổ
1
t ữủ ự õ ởt số ỳ t
2
ũ s ữớ t t r õ ởt
ừ tr t số ỳ t
r ố t õ t ồ
ụ ữ sữ ự ữỡ tr tự
ữỡ tr õ t tự ữớ t t
ự ử ú ừ ỹ tr tt



◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

♥❤❛✉ tø ✈➟t ❧➼✱ ❤â❛ ❤å❝✱ ❦➽ t❤✉➟t✱ s✐♥❤ ❤å❝ ✤➳♥ t➔✐ ❝❤➼♥❤✱✳✳✳✈✳✈✳ ❚✉②
♥❤✐➯♥✱ ❝➙✉ ❤ä✐ ❝ì ❜↔♥ ✧❑❤✐ ♥➔♦ ♠ët ❤➔♠ ❝â ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù❄✧✱ ✈➝♥
❝❤÷❛ ❝â ❝➙✉ tr↔ ❧í✐ ✤➛② ✤õ✳ ●➛♥ ✤➙②✱ tr♦♥❣ ❜➔✐ ❜→♦

✧❲❤✐❝❤ ❢✉♥❝t✐♦♥s

❛r❡ ❢r❛❝t✐♦♥❛❧❧② ❞✐❢❢❡r❡♥t✐❛❜❧❡❄✧✱ t→❝ ❣✐↔ ●✳❱❛✐♥✐❦❦♦ ❝â ✤÷❛ r❛ ❝→❝ t➼♥❤
❝❤➜t ✤➸ ❦✐➸♠ tr❛ ❦❤✐ ♥➔♦ ♠ët ❤➔♠ ❧➔ ❦❤↔ t➼❝❤ ♣❤➙♥ t❤ù❄ ▲✉➟♥ ✈➠♥ ♥➔②
✤÷ñ❝ ❞ò♥❣ ✤➸ ♣❤➙♥ t➼❝❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❜➔✐ ❜→♦ ♥â✐ tr➯♥✳
▲✉➟♥ ✈➠♥ ❣ç♠ ❤❛✐ ❝❤÷ì♥❣✿
❈❤÷ì♥❣ ✶✿ ●✐î✐ t❤✐➺✉ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝ì ❜↔♥ ❧✐➯♥ q✉❛♥
✤➳♥ ♣❤➨♣ t➼♥❤ ✈✐✲t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù✳
❈❤÷ì♥❣ ✷✿ ●✐î✐ t❤✐➺✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t✐➯✉ ❝❤✉➞♥ ✤➸ ♠ët ❤➔♠ ❦❤↔
t➼❝❤ ♣❤➙♥ t❤ù ✭❜❛♦ ❣ç♠ ❦❤↔ t➼❝❤ t❤❡♦ ♥❣❤➽❛ ♥❣✉②➯♥ t❤õ② ✈➔ t❤❡♦ ♥❣❤➽❛
❈❛♣✉t♦✮✳

✸


❈❤÷ì♥❣ ✶
▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❣✐↔✐
t➼❝❤ ♣❤➙♥ t❤ù
✶✳✶ ❚➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡
❚r♦♥❣ ♠ö❝ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ ❣✐î✐ t❤✐➺✉ t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù ❘✐❡♠❛♥♥✲
▲✐♦✉✈✐❧❧❡ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♥â✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ α ∈ R+, x : [a, b] → R✱ ❦❤✐ ✤â t➼❝❤ ♣❤➙♥ ♣❤➙♥

t❤ù ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❝➜♣ α ❝õ❛ ❤➔♠ x ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐
1
J x(t) :=
Γ(α)

t

α

(t − τ )α−1 x(τ )dτ,

a ≤ t ≤ b.

a

❱î✐ α = 0✱ ❝❤ó♥❣ t❛ q✉② ÷î❝ J 0 := I ❧➔ t♦→♥ tû ✤ç♥❣ ♥❤➜t✳
❙❛✉ ✤➙②✱ ❝❤ó♥❣ t❛ ❝❤♦ ✤✐➲✉ ❦✐➺♥ ✤➸ ♠ët ❤➔♠ ❝â t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù
❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡✳
❑➼ ❤✐➺✉ L1 [a, b] ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ tr➯♥ [a, b] ✈➔ C[a, b] ❧➔
❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ [a, b].

✣à♥❤ ❧þ ✶✳✶✳ ❈❤♦ f ∈ L1[a, b] ✈➔ α > 0✱ ❦❤✐ ✤â t➼❝❤ ♣❤➙♥ J αf (x) tç♥
t↕✐ ❤➛✉ ❤➳t x ∈ [a, b]✳ ❍ì♥ ♥ú❛✱ ❤➔♠ J αf ❝ô♥❣ ❦❤↔ t➼❝❤ tr➯♥ [a, b]✳
✹


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❈❤ù♥❣ ♠✐♥❤✳ ❳❡♠ tr♦♥❣ ❬✶✱ ❚❤❡♦r❡♠ ✷✳✶❪
▼ët t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ❝→❝ t♦→♥ tû t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù ❧➔
t➼♥❤ ❝❤➜t ♥û❛ ♥❤â♠✳

✣à♥❤ ❧þ ✶✳✷✳ ❈❤♦ m, n ≥ 0, φ ∈ L1[a, b] t❤➻
J m J n φ = J m+n φ

✤ó♥❣ ❤➛✉ ❦❤➢♣ ♥ì✐ tr➯♥ [a, b]✳ ❚❤➯♠ ♥ú❛✱ φ ∈ C[a, b] ❤♦➦❝ m + n ≥
1, n ≥ 1 t❤➻ ✤✐➲✉ ♥➔② ✤ó♥❣ ❦❤➢♣ ♥ì✐ tr➯♥ [a, b]✳
❈❤ù♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ ❝â
J m J n φ(x) =

1
Γ(m)Γ(n)

x

t

(x − t)m−1

(t − τ )n−1 φ(τ )dτ dt.

a

a

❙û ❞ö♥❣ ✣à♥❤ ❧➼ ❋✉❜✐♥✐ ✤➸ t❤❛② ✤ê✐ t❤ù tü t➼♥❤ t➼❝❤ ♣❤➙♥ t❛ t❤✉ ✤÷ñ❝

1

J J φ(x) =
Γ(m)Γ(n)
1
=
Γ(m)Γ(n)

x

x

(x − t)m−1 (t − τ )n−1 φ(τ )dτ dt

m n

τ

a
x

x

(x − t)m−1 (t − τ )n−1 dtdτ.

φ(τ )
a

τ

✣ê✐ ❜✐➳♥ t = τ + s(x − τ ) ❝❤ó♥❣ t❛ ❝â
x

1
J J =
Γ(m)Γ(n)
1
=
Γ(m)Γ(n)
m n

1

[(x − τ )(1 − s)]m−1 × [s(x − τ )]n−1 (x − τ )dsdτ

φ(τ )
a

0
x

1
m+n−1

(1 − s)m−1 sn−1 dsdτ.

φ(τ )(x − τ )
a

0

◆❣♦➔✐ r❛✱ ❜➡♥❣ t➼♥❤ t♦→♥ trü❝ t✐➳♣ t❛ ❝â✿

1

(1 − s)m−1 sn−1 ds =
0

✺

Γ(m)Γ(n)
.
Γ(m + n)


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❉♦ ✤â

1
J J φ(x) =
Γ(m + n)

x

m n

φ(τ )(x − τ )m+n−1 dτ = J m+n φ(x)
a

❤➛✉ ❦❤➢♣ ♥ì✐ tr➯♥ [a, b]✳
❍ì♥ ♥ú❛✱ t❤❡♦ ❝→❝ ❦➳t q✉↔ ✈➲ t➼❝❤ ♣❤➙♥ ♣❤ö t❤✉ë❝ t❤❛♠ sè✱ ♥➳✉

φ ∈ C[a, b] t❤➻ J n φ ∈ C[a, b]✱ ✈➔ ❞♦ ✤â ❝❤ó♥❣ t❛ ❝ô♥❣ ❝â Jam Jan φ ∈
C[a, b], Jam+n φ ∈ C[a, b]. ✣✐➲✉ ♥➔② ❞➝♥ tî✐ J m J n φ(x) = J m+n φ(x), ∀x ∈
[a, b].
❈✉è✐ ❝ò♥❣✱ ♥➳✉ φ ∈ L1 [a, b] ✈➔ m + n ≥ 1, n ≥ 1 t❤❡♦ ❦➳t q✉↔ tr➯♥ t❛
❝â

J m J n φ = J m+n φ = J m+n−1 J 1 φ
❤➛✉ ❦❤➢♣ ♥ì✐✳ ❇ð✐ J 1 ❧✐➯♥ tö❝ ♥➯♥ t❛ ❝â J m+n φ = J m+n−1 J 1 φ ❝ô♥❣ ❧✐➯♥
tö❝✳ ❉♦ ✤â ❝❤ó♥❣ t❛ t❤✉ ✤÷ñ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
✣➦t✿

H µ [a, b] := {f : [a, b] → R; ∃c > 0, ∀x, y ∈ [a, b] : |f (x) − f (y)| ≤ |x − y|µ } ,
H ∗ [a, b] := f : [a, b] → R : ∃L > 0 : |f (x + h) − f (x)| ≤ L|h|ln|h|−1 .
❈❤ó♥❣ t❛ ❝ò♥❣ t❤❡♦ ❞ã✐ ✣à♥❤ ❧➼ s❛✉✿

✣à♥❤ ❧þ ✶✳✸✳ ❈❤♦ φ ∈ H µ[a, b] ✈î✐ µ ∈ [0, 1]✱ ✈➔ ❝❤♦ 0 < n < 1✳ ❚❤➳
t❤➻

J n φ(x) =

φ(a)
(x − a)n + ϕ(x).
Γ(n + 1)

✻


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

Ð ✤➙②✱ ❤➔♠ ϕ t❤ä❛ ♠➣♥
ϕ(x) = O((x − a)µ+n )

❦❤✐ x → a✳
◆❣♦➔✐ r❛





H µ+n [a, b],



ϕ ∈ H ∗ [a, b],





H 1 [a, b],

µ + n < 1,
µ + n = 1,
µ + n > 1.

❈❤ù♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ ❝â
x

φ(a)
J φ(x) =
Γ(n)
n

n−1

(x − t)
a

1
dt +
Γ(n)

x
a

φ(t) − φ(a)
dt.
(x − t)1−n

✣✐➲✉ ♥➔② ❞➝♥ tî✐

1
ϕ(x) =
Γ(n)

x
a

φ(t) − φ(a)
dt.
(x − t)1−n

❙û ❞ö♥❣ ❣✐↔ t❤✐➳t φ ∈ H µ ✱ ❝❤ó♥❣ t❛ ❝â
x
x
1
L|t − a|µ
L
|ϕ(x)| ≤
dt =
(t − a)µ (x − t)n−1 dt
1−n
Γ(n) a (x − t)
Γ(n) a
1
L
=
(x − a)µ+n
sµ (1 − s)n−1 ds = O((x − a)µ+n ).
Γ(n)
0

❇➙② ❣✐í✱ ✤➦t g(x) :=

φ(x) − φ(a)
✈➔ ❧➜② h > 0 s❛♦ ❝❤♦ x, x + h ∈ [a, b]✳
Γ(n)

❈❤ó♥❣ t❛ ❝â
x+h

x
n−1

ϕ(x + h) − ϕ(x) =

g(t)(x + h − t)
a

g(t)(x − t)n−1 dt

dt −
a

✼


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
x

g(t)[(x + h − t)n−1 − (x − t)n−1 ]dt

=
a

x+h

g(t)(x + h − t)n−1 dt

+
x

= K1 + K2 + K3 .
❚r♦♥❣ ✤â
x+h

(g(t) − g(x))[(x + h − t)n−1 − (x − t)n−1 ]dt,

K1 :=
a
x+h

(g(t) − g(x))(x + h − t)n−1 dt,

K2 :=
x

x

x+h
n−1

[(x + h − t)

K3 := g(x)

n−1

− (x − t)

(x + h − t)n−1 dt .

]dt +

a

x

❙❛✉ ✤➙②✱ ❝❤ó♥❣ t❛ s➩ ÷î❝ ❧÷ñ♥❣ ❝→❝ ❤↕♥❣ tû K1 , K2 , K3 ✳ ❉♦ g ∈ H µ ✱
t❛ ❝â✿
x−a

(g(x − u) − g(x))[(u + h)n−1 − un−1 ]du|

|K1 | = |
0

x−a

uµ [un−1 − (u + h)n−1 ]du

≤L
0

(x−a)/h

(ht)µ [(th)n−1 − (th + h)n−1 ]dt

= Lh
0

(x−a)/h
µ+n

tµ [tn−1 − (t + 1)n−1 ]dt.

= Lh

0

❚r♦♥❣ tr÷í♥❣ ❤ñ♣ x−a < h✱ t➼❝❤ ♣❤➙♥ ❜à ❝❤➦♥ ❜ð✐

1 µ+n−1
dt
0 t

✈➔ ❞♦ ✤â K1 = O(hµ+n )✳ ◆➳✉ x − a ≥ h✱ ❝❤ó♥❣ t❛ t❤➜② r➡♥❣
(x−a)/h

tµ [tn−1 − (t + 1)n−1 ]dt
0

✽

=

1
✱
µ+n


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
1

(x−a)/h
µ n−1

=

t [t

− (t + 1)

n−1

0

<

tµ [tn−1 − (t + 1)n−1 ]dt

]dt +
1

(x−a)/h

1
+ (1 − n)
µ+n

tµ+n−2 dt.
1

❈❤ó♥❣ t❛ t❤➜② ✈î✐ µ + n < 1 t❤➻

|K1 | ≤ O(h
≤ O(h

µ+n

µ+n

)

1
+ (1 − n)
µ+n

1
)
+ (1 − n)
µ+n

(x−a)/h

tµ+n−2 dt
1
∞

tµ+n−2 dt

≤ O(hµ+n ),

1

❜ð✐ ✈➻ µ + n < 1 ✈➔ x − a ≤ h✳ ❱î✐ µ + n = 1 t➼♥❤ t♦→♥ ♠ët ❝→❝❤ t÷ì♥❣
tü t❛ t❤✉ ✤÷ñ❝
(x−a)/h
µ+n

|K1 | ≤ O(h

t−1 dt

) 1+

= O(h ln h−1 ).

1

❈✉è✐ ❝ò♥❣✱ ✈î✐ µ + n > 1 ❝❤ó♥❣ t❛ ❝â✿ ❜ð✐ ✈➻ x ≤ b✱
(b−a)/h
µ+n

|K1 | ≤ O(h

µ+n−2

)

t

µ+n

dt = O(h

1

❉♦ ✤â

b−a
)
h

µ+n−1

= O(h).





O(hµ+n ),

µ + n < 1,



K1 = O(h ln h−1 ),
µ + n = 1,





O(h),
µ + n > 1.

❚✐➳♣ t❤❡♦✱ t❛ ÷î❝ ❧÷ñ♥❣ K2 ✳ ❈❤ó♥❣ t❛ sû ❞ö♥❣ ♠ët ❧➛♥ ♥ú❛ ✤✐➲✉ ❦✐➺♥

g ∈ H µ ✤➸ s✉② r❛ ✭ sû ❞ö♥❣ ♣❤➨♣ t❤➳ s = (t − x)/h✮
x+h

(t − x)µ (x + h − t)n−1 dt

|K2 | ≤ L
x

✾


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

1
µ+n

sµ (1 − s)n−1 ds = O(hµ+n ).

= Lh

0

❈✉è✐ ❝ò♥❣✱ ✈î✐ K3 ✱ ❝❤ó♥❣ t❛ sû ❞ö♥❣ ❣✐↔ t❤✐➳t ❧✐➯♥ tö❝ ❍☎
♦❧❞❡r ❝õ❛ φ✱
s✉② r❛ |g(x)| ≤ L(x − a)µ ✈î✐ ❤➡♥❣ sè ▲ ♥➔♦ ✤â✳ ×î❝ ❧÷ñ♥❣ trü❝ t✐➳♣
❝→❝ t➼❝❤ ♣❤➙♥ ♥➔②✱ ❝❤ó♥❣ t❛ ❝â✿

|K3 | ≤

L
(x − a)µ [(x − a + h)n − (x − a)n ].
n

❚r♦♥❣ tr÷í♥❣ ❤ñ♣ x − a ≤ h✱ ❜✐➸✉ t❤ù❝ tr➯♥ ❜à ❝❤➦♥ tr➯♥ ❜ð✐ O(hµ+n )✳
◆➳✉ x − a > h✱ t❛ ÷î❝ ❧÷ñ♥❣ ❤↕♥❣ tû tr♦♥❣ ♥❣♦➦❝ t❤❡♦ ✣à♥❤ ❧➼ ❣✐→ trà
tr✉♥❣ ❜➻♥❤ ❝õ❛ ♣❤➨♣ ❧➜② ✈✐ ♣❤➙♥ ✈➔ t❤➜② ✭❝â t➼♥❤ ✤➳♥ ♥ ❁✶✮

K3 = O(1)(x − a)µ h(x − a)n−1 = O(h)(x − a)µ+n−1 .
❑➳t ❤ñ♣ t➜t ❝↔ ❝→❝ ÷î❝ ❧÷ñ♥❣ ♥➔② t❛ ✤÷ñ❝






O(hµ+n ),
µ + n < 1,



ϕ(x + h) − ϕ(x) = O(h ln h−1 ),
µ + n = 1,





O(h),
µ + n > 1.

✶✵


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

✶✳✷ ✣↕♦ ❤➔♠ ♣❤➙♥ t❤ù
✶✳✷✳✶ ✣↕♦ ❤➔♠ ♣❤➙♥ t❤ù ♥❣✉②➯♥ t❤õ②
❳➨t t♦→♥ tû t➼❝❤ ♣❤➙♥ ❘✐❡♠❛♥♥✕▲✐♦✉✈✐❧❧❡ J α : C[0, T ] → C[0, T ] ✈î✐
❝➜♣ α ∈ R+ ①→❝ ✤à♥❤ ❜ð✐

1
(J u)(t) =

Γ(α)

t

α

(t − s)α−1 u(s)ds, 0 ≤ t ≤ T, u ∈ C[0, T ]
0

♥❤÷ ✤➣ ✤÷ñ❝ ❣✐î✐ t❤✐➺✉ ð tr♦♥❣ ♠ö❝ ✶✳✶✳ ◆â✐ r✐➯♥❣✱ (J 1 u)(t) =

t
0 u(s)ds

✈î✐ α = m ∈ N = {1, 2, ...} , ❣✐→ ❝õ❛ t♦→♥ tû J m ✤÷ñ❝ ❝❤♦ ❜ð✐

J m C [0, T ] = v ∈ C m [0, T ] : v (k) (0) = 0, k = 0, ..., m − 1 =: C0m [0, T ] ,
m

d
✈➔ J ❦❤↔ ♥❣❤à❝❤ tr➯♥ ✤â✱ (J ) v =
tr♦♥❣ ✤â
=
:
dt
C m [0, T ] → C [0, T ]✳ ❉♦ t➼♥❤ ❝❤➜t ♥û❛ ♥❤â♠ ❝õ❛ t♦→♥ tû t➼❝❤ ♣❤➙♥ t❛
m −1

m

D0m v ✱

D0m

❝â✿

J α J β = J α+β : α > 0, β > 0,

✭✶✳✶✮

①❡♠ tr♦♥❣ ✣à♥❤ ❧➼ ✶✳✷✳
❚♦→♥ tû J α ❧➔ ❦❤↔ ♥❣❤à❝❤ tr➯♥ J α C [0, T ] ✈î✐ ❝➜♣ α ❦❤æ♥❣ ♥❣✉②➯♥✳
❚❤➟t ✈➟②✱ ♥➳✉ J α u = 0 ✈î✐ u ∈ C [0, T ] t❤➻ t❛ ❧➜② m ∈ N, m > α✱
❝❤ó♥❣ t❛ ❝â J m u = J m−α J α u = 0. ✣✐➲✉ ♥➔② ❞➝♥ tî✐ u = 0✳ ❙ü ♠✐➯✉
t↔ ❝õ❛ ❣✐→ J α C [0, T ] , α > 0 ❝â ❧✐➯♥ q✉❛♥ ❝❤➦t ❝❤➩ tî✐ sü ♠✐➯✉ t↔ ❝õ❛
❝→❝ ❤➔♠ ❦❤↔ ✈✐ ♣❤➙♥ t❤ù✳ ❈ö t❤➸✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ t♦→♥ tû
✤↕♦ ❤➔♠ ♥❣✉②➯♥ t❤õ② ❝➜♣ α ❝õ❛ ❝→❝ ❤➔♠ tr♦♥❣ J α C[0, T ]✱

D0α v = (J α )−1 v, v ∈ J α C [0, T ] .
✶✶

✭✶✳✷✮


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❑❤✐ ✤↕♦ ❤➔♠ ♥➔② tç♥ t↕✐✱ ❝❤ó♥❣ t❛ ❝â t➼♥❤ ❝❤➜t ♥û❛ ♥❤â♠ ♥❤÷ s❛✉✿

D0α D0β = D0β D0α = D0α+β , α > 0, β > 0.

✭✶✳✸✮

❚❤❡♦ ✭✶✳✷✮✱ ♠ët ❤➔♠ v ∈ C [0, T ] ❧➔ D0α ✲❦❤↔ ✈✐ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♣❤÷ì♥❣
tr➻♥❤ J α u = v ❝â ♠ët ♥❣❤✐➺♠ u ∈ C [0, T ] . ❱î✐ m < α < m + 1, m ∈

N0 = {0, 1, 2, ...} , ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t÷ì♥❣ ✤÷ì♥❣ ✈î✐ J m+1−α J α u =
J m+1−α v ✱ ❤♦➦❝ ✈î✐✱ J m+1 u = J m+1−α v ✱ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❝â t❤➸ ❣✐↔✐
✤÷ñ❝ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ J m+1−α v ∈ C0m+1 [0, T ] ; ♥❣❤✐➺♠ ✤÷ñ❝ ❝❤♦ ❜ð✐

u = D0α v = Dm=1 J m+1−α v ✳ ❉♦ ✤â✱ m < α < m + 1, m ∈ N0 ✱ ❝❤ó♥❣ t❛
❝â✿

D0α v = Dm+1 J m+1−α v,

✭✶✳✹✮

✈î✐ v ∈ C [0, T ] s❛♦ ❝❤♦ J m+1−α v ∈ C0m+1 [0, T ] .
✣✐➲✉ ♥➔② ❝â t❤➸ ✤÷ñ❝ ①❡♠ ①➨t ♥❤÷ ♠ët ✤à♥❤ ♥❣❤➽❛ ❦❤→❝ ❝õ❛ D0α ✳

✶✳✷✳✷ ✣↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦
❇➯♥ ❝↕♥❤ ❦❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ♥❣✉②➯♥ t❤õ② ❝â ♥❤✐➲✉
❦❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❦❤→❝ ✤➣ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛✳ ❚r♦♥❣ sè ♥➔②✱
✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦ ✤÷ñ❝ sû ❞ö♥❣ rë♥❣ r➣✐ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥
❧➼ t❤✉②➳t ✈➔ ù♥❣ ❞ö♥❣✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ♥❣❤✐➯♥ ❝ù✉
❝→❝ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ♠ët ❤➔♠ ❝â ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦✳
❈❤♦ α > 0, x : [a, b] → R, ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦ ❝➜♣ α ❝õ❛ x
✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐
α

DCap
x(t) := D[α+1] J [α+1]−α x(t), t ∈ (a, b] .

✶✷


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

0
:= I ❧➔ t♦→♥ tû ✤ç♥❣
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ α = 0✱ ❝❤ó♥❣ t❛ q✉② ÷î❝ DCap

♥❤➜t✳
❉➵ t❤➜② ♥➳✉ x ∈ C [α+1] [a, b] t❤➻ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦ ❝➜♣ α tç♥
t↕✐✳ ❙❛✉ ✤➙②✱ ❝❤ó♥❣ t❛ s➩ ❣✐î✐ t❤✐➺✉ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ t➼❝❤ ♣❤➙♥ ♣❤➙♥
t❤ù ✈➔ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦✳

✣à♥❤ ❧þ ✶✳✹✳ ◆➳✉ f ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✈➔ α ≥ 0✱ t❤➻
α
DCap
J α f = f.

❈❤ù♥❣ ♠✐♥❤✳ ❳❡♠ tr♦♥❣ ❬✶✱ ❚❤❡♦r❡♠ ✸✳✼❪
◆❣♦➔✐ r❛✱ ❦❤→❝ ✈î✐ ♣❤➨♣ t➼♥❤ ✈✐✲t➼❝❤ ♣❤➙♥ ❝ê ✤✐➸♥✱ ❝❤ó♥❣ t❛ ❝ô♥❣
t❤➜② r➡♥❣ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦ ❦❤æ♥❣ ❧➔ ♥❣❤à❝❤ ✤↔♦ ♣❤↔✐ ❝õ❛
t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡✳

✣à♥❤ ❧þ ✶✳✺✳ ●✐↔ sû α ≤ 0, m = [α + 1] ✈➔ f ∈ C m[a, b] t❤➻

m−1

J

α

α
DCap
f (x)

Dk f.

= f (x) −
k=0

❈❤ù♥❣ ♠✐♥❤✳ ❳❡♠ tr♦♥❣ ❬✶✱ ❚❤❡♦r❡♠ ✸✳✽❪✳

✶✸


❈❤÷ì♥❣ ✷
❑❤✐ ♥➔♦ ♠ët ❤➔♠ ❝â ✤↕♦ ❤➔♠
♣❤➙♥ t❤ù❄
✷✳✶ P❤→t ❜✐➸✉ ❦➳t q✉↔ ❝❤➼♥❤
◆❤➢❝ ❧↕✐ r➡♥❣ H α [0, T ] , 0 < α ≤ 1✱ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝
❍☎♦❧❞❡r ✈î✐ ❝❤✉➞♥ ♥❤÷ s❛✉

v

H α :=

max | v (t) | + sup | v (t) − v (s)|(t − s)α α < ∞,

0≤t≤T

0≤s≤t≤T

✈➔ H0α [0, T ] , 0 < α < 1, ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣ ❝õ❛ H α [0, T ] ❝❤ù❛ ❝→❝
❤➔♠ v ∈ H α [0, T ] t❤ä❛ ♠➣♥

| v (t) − v (s)|
→ 0,
(t − s)α
0≤s≤t≤T,t−s≤ε
sup

ε → 0.

◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ ❝❤ù♥❣ ♠✐♥❤ ✣à♥❤ ❧➼ s❛✉✳

✣à♥❤ ❧þ ✷✳✶✳ ❱î✐ α ∈ (0, 1) ✈➔ v ∈ C [0, T ]✱❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ❧➔
t÷ì♥❣ ✤÷ì♥❣✳

✭✐✮ v ∈ J αC [0, T ]✱ tù❝ ❧➔ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù D0α := (J α) v ∈ C [0, T ]
✶✹


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

tç♥ t↕✐✳
✭✐✐✮ ●✐î✐ ❤↕♥ γ0 := t−→0
lim t−α v (t) tç♥ t↕✐✱ ✈➔ t➼❝❤ ♣❤➙♥
t
−α−1
(v (t) − v (s)) ds, 0 < t ≤ T ❤ë✐ tö t❤❡♦ ♥❣❤➽❛
0 (t − s)
t

lim sup |
θ↑1 0≤t≤T
θt

(t − s)−α−1 (v (t) − v (s)) ds

0

(t − s)−α−1 (v (t) − v (s)) ds| = 0.

−

✭✷✳✶✮

0

✭✐✐✬✮ ●✐î✐ ❤↕♥ γ0 := lim
t−α v (t) tç♥ t↕✐❀ t➼❝❤ ♣❤➙♥
t→0
t

−α−1
((vt) − v (s)) ds =: ω (t) ❤ë✐ tö ✈î✐ t ∈ (0, T ] ✈➔ ①→❝
0 (t − s)
✤à♥❤ ♠ët ❤➔♠ ω ∈ C (0, T ]❝â ♠ët ❣✐î✐ ❤↕♥ ❤ú✉ ❤↕♥ ❦❤✐ t → 0❀
❤ì♥ ♥ú❛✱ ❝â ♠ët ❤➔♠ W ∈ L1 (0, T ) s❛♦ ❝❤♦
θt

|

(t − s)−α−1 (v (t) − v (s)) ds| ≤ W (t) , 0 < t < T, 0 < θ < 1.

0

✭✐✐✐✮ v ❝â ❝➜✉ tró❝

✭✷✳✷✮

tr♦♥❣ ✤â
H0α [0, T ] , v0 (0) = 0✱ ✈➔ t➼❝❤ ♣❤➙♥
v = γ0 tα + v0

t

γ0

❧➔ ♠ët ❤➡♥❣ sè✱

v0 ∈

(t − s)−α−1 (v (t) − v (s)) ds := ω (t)

0

❤ë✐ tö ✈î✐ t ∈ (0, T ] ✈➔ ①→❝ ✤à♥❤ ♠ët ❤➔♠ ω ∈ C (0, T ] ♠➔ ❝â ♠ët
❣✐î✐ ❤↕♥ ❤ú✉ ❤↕♥ ω (0) := lim
ω (t) ✭✈➻ ✈➟② ω ∈ C [0, T ]✮✳
t→0
✭✐✐✐✬✮ v ❝â ❝➜✉ tró❝ v

= γ0 tα + v0 ✱

tr♦♥❣ ✤â γ0 ❧➔ ♠ët ❤➡♥❣ sè✱ v0 ∈
H0α [0, T ] , v (0) = 0✱ ✈➔ t➼❝❤ ♣❤➙♥
t
−α−1
(v (t) − v (s)) ds := ω (t) ❤ë✐ tö ✈î✐ t ∈ (0, T ] ✈➔ ①→❝
0 (t − s)
✤à♥❤ ♠ët ❤➔♠ ω0 (0) = 0 ✈î✐ ω0 ∈ C [0, T ]✳ ❱î✐ v ∈ J αC [0, T ]✱
✶✺


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
0
t❛ ❝â✿

(D0α v) (t) := (J α )−1 v (t)
1

t−α v (t) + α
=
Γ (1 − α)

t

(t − s)−α−1 (v (t) − v (s)) ds ;

0

✭✷✳✸✮

(D0α v) (0) := (J α )−1 v (0) = Γ (α + 1) γ0 .
❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝õ❛ ✤à♥❤ ❧➼ ♥➔② s➩ ✤÷ñ❝ tr➻♥❤ ❜➔② ð ❝✉è✐ ❝❤÷ì♥❣✳
❚r÷î❝ ❦❤✐ ✤✐ ✈➔♦ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧➼ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ ❣✐î✐ t❤✐➺✉ ♠ët
sè ❤➔♠ ❦❤↔ t➼❝❤ ♣❤➙♥ t❤ù✳

✷✳✷ ▼ët sè ❤➔♠ ❝â ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù
✷✳✷✳✶ ❳➨t ❤➔♠ v ∈ H β [0, T ] , 0 < α < β ≤ 1, v (0) = 0.
❍➔♠ v ♥➔② t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (ii) ❝õ❛ ✣à♥❤ ❧➼ ✷✳✶✱ ✈➻ ✈➟② D0α v ∈

C [0, T ] ✤÷ñ❝ ①→❝ ✤à♥❤ ❜➡♥❣ ❝æ♥❣ t❤ù❝ ✭✷✳✸✮✳ ❙❛✉ ✤➙②✱ ❝❤ó♥❣ t❛ s➩ ❦✐➸♠
tr❛ t➼♥❤ ❦❤↔ t➼❝❤ ♣❤➙♥ t❤ù ❝õ❛ ❝→❝ ❤➔♠ t❤✉ë❝ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ s❛✉✿

W01,p (0, T ) = v ∈ C[0, T ] : v ∈ Lp (0, T ), v(0) = 0.

▼➺♥❤ ✤➲ ✷✳✶✳ ❱î✐ 0 < α < 1, p > 1 −1 α ✱ ❦❤✐ ✤â✿
W01,p (0, T ) ⊂ J α [0, T ],

✈î✐ v ∈ W01,p (0, T ) .

❈❤ù♥❣ ♠✐♥❤✳ ❇➜t ✤➥♥❣ t❤ù❝ p > 1 −1 α

D0α v = J 1−α v .

✭✷✳✹✮

p−1
> α✳ ❱î✐ v ∈
p
W01,p (0, T ) , 0 ≤ s < t ≤ T, t − s ≤ 1✱ sû ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍☎♦❧❞❡r
✶✻

❦➨♦ t❤❡♦


◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
t❛ ❝â✿
t

|v (t) − v (s)| = |

v (τ ) dτ | ≤ v

Lp

p−1
(t − s) p ,

s

p−1
> α✳ ❱➻ t❤➳✱ v ∈ J α C [0, T ] ✈➔ ✭✷✳✸✮
p
❧➔ ✤ó♥❣✳ ❚ø ✭✷✳✸✮ t❛ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ✭✷✳✸✮ ❞➝♥ tî✐ D0α v = J 1−α v ✳
❞♦ ✤â✱ v ∈ H β [0, T ] , β =

❱➻ ✈➟②✱ ✭✷✳✹✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳

✷✳✷✳✷ ❳➨t ❤➔♠ tα, α > 0
1
tα ✱ ❤➔♠ tα ❧➔ D0α ✲ ❦❤↔ ✈✐✳ ❚❤❡♦ ✤à♥❤ ❧➼ ✷✳✶
Γ (α + 1)
tα ✈î✐ 0 < α < 1 t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (ii)✳ ❈❤ó♥❣ t❛ s➩ ❦✐➸♠ tr❛ trü❝
❇ð✐ ✈➻ (J α 1) (t) =

t✐➳♣ ❦❤➥♥❣ ✤à♥❤ ♥➔②✳ ❚❤➟t ✈➟②✿ sû ❞ö♥❣ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ✈➔ ♣❤➨♣
✤ê✐ ❜✐➳♥ s = tx✿
t
−α−1

(t − s)
θt

t
tα − s α t
(t − s )ds =
(t − s)−α sα−1 ds
α |s=θt +

α (t − s)
θt
1
α
1−θ
=−
(1 − x)−α xα−1 dx → 0.
α +
α (1 − θ)
θ
α

α

✭✷✳✺✮
❉♦ ✤â ✤✐➲✉ ❦✐➺♥ (ii) ✤÷ñ❝ ❦✐➸♠ tr❛✳

✷✳✸ ✣↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦
❱î✐ v ∈ C[0, T ] ✈➔ J 1−α (v − v(0)) ∈ C 1 [0, T ]✱ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù
❈❛♣✉t♦ ❜➟❝ α ❝õ❛ v ✤÷ì❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐
α
α
DCap
v = D1 J 1−α (v − v(0)) = DR−L
(v − v(0)).

✶✼

✭✷✳✻✮

◆❣✉②➵♥ ◆❣å❝ ❍✉②➲♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

▼è✐ q✉❛♥ ❤➺ ❣✐ú❛ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦ ✈➔ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù
♥❣✉②➯♥ t❤õ② ✤÷ñ❝ t❤➸ ❤✐➺♥ tr♦♥❣ ▼➺♥❤ ✤➲ s❛✉✿

▼➺♥❤ ✤➲ ✷✳✷✳ ▼ët ❤➔♠ v ∈ C[0, T ] ❝â ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦

α
v ∈ C[0, T ], 0 < α < 1✱
DCap

♥➳✉ ✈➔ ❝❤➾ ♥➳✉ v − v(0) ❝â ✤↕♦ ❤➔♠ ♣❤➙♥
α
v = D0α (v − v(0))✳
t❤ù D0α(v − v(0)) ∈ C[0, T ]✳ ❍ì♥ ♥ú❛✱ DCap

❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ 0 < α < 1✱ ✤✐➲✉ ❦✐➺♥ v ∈ C[0, T ], v(0) = 0, ❞➝♥ tî✐
(J 1−α v)(0) = 0✳ ❱➻ ✈➟② ❣✐↔ t❤✐➳t J 1−α (v − v(0)) ∈ C 1 [0, T ] tr♦♥❣ ✭✷✳✻✮
t÷ì♥❣ ✤÷ì♥❣ ✈î✐ J 1−α (v − v(0)) ∈ C01 [0, T ]✳ ❱➻ ✈➟②✱ D0α (v − v(0)) ✤÷ñ❝
α
①→❝ ✤à♥❤ ✈➔ DCap
v = D1 J 1−α (v − v(0)) = D0α (v − v(0))✳

❑➳t ❤ñ♣ ▼➺♥❤ ✤➲ tr➯♥ ✈î✐ ✣à♥❤ ❧➼ ✷✳✶ ❝❤ó♥❣ t❛ t❤✉ ✤÷ñ❝ ♥❣❛② ❦➳t
q✉↔ s❛✉✳

✣à♥❤ ❧þ ✷✳✷✳ ❱î✐ 0 < α < 1 ✈➔ v ∈ C[0, T ]✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❞÷î✐ ✤➙② ❧➔

t÷ì♥❣ ✤÷ì♥❣✳

α
✭✐✮ ✣↕♦ ❤➔♠ ♣❤➙♥ t❤ù DCap
v ∈ C[0, T ] tç♥ t↕✐✳

✭✐✐✮ ●✐î✐ ❤↕♥ lim
t−α (v(t) − v(0)) =: γ tç♥ t↕✐✱ ✈➔
t→0
t

(t − s)−α−1 (v(t) − v(s))ds| → 0,

sup |
0≤t≤T

θ ↑ 1.

θt

✭✐✐✐✮ v ❝â ❝➜✉ tró❝ v − v(0) = γtα + v0 tr♦♥❣ ✤â γ ❧➔ ♠ët ❤➡♥❣ sè✱
t
v0 ∈ H0α [0, T ] ✱ ✈➔ 0 (t − s)−α−1 (v(t) − v(s))ds =: ω0 (t) ❤ë✐ tö ✈î✐
♠é✐ t ∈ (0, T ] ①→❝ ✤à♥❤ ♠ët ❤➔♠ ω0 ∈ C (0, T ] ❝â ❣✐î✐ ❤↕♥ ❤ú✉
❤↕♥ lim
ω0 (t) =: ω0 (0).
t→0
α
α

❱î✐ v ∈ C[0, T ] ✈➔ DCap
v ∈ C[0, T ] t❤➻ (DCap
v)(0) = Γ(α + 1)γ,
✶✽