ớ t tổ tọ ỏ t ỡ s s tợ ừ ỵ
t ổ tr ở ổ t t t ụ t ũ ữớ
ữợ ự ữợ ụ ữ ở tổ õ t ỹ t õ
tổ ỷ ớ ỡ tợ t t ợ P õ
ỳ ỵ õ õ ú ù ở t t ủ tổ t õ


ổ tr t ỡ

ỡ t
tỹ






▼ö❝ ❧ö❝
▼ð ✤➛✉
✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✶✳✶ ❑❤→✐ ♥✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ✈î✐ ♥❤➙♥ t→❝❤ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸
✹
✹
✺
✽

✷ P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆ ❋❘❊❉❍❖▲▼ ▲❖❸■ ❍❆■ ❱❰■ ◆❍❹◆
❚✃◆● ◗❯⑩❚
✶✷
✷✳✶
✷✳✷
✷✳✸
✷✳✹

P❤÷ì♥❣ ♣❤→♣ t❤➳ ❧✐➯♥ ❤ñ♣ ✳ ✳
P❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣
❈→❝ ✤à♥❤ ❧þ ❋r❡❞❤♦❧♠ ✳ ✳ ✳ ✳
❈➜✉ tró❝ ❝õ❛ ♥❤➙♥ ❣✐↔✐ ✳ ✳ ✳ ✳

✳
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✳
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✶✷
✶✺
✶✽
✷✺

✸ P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆ ❋❘❊❉❍❖▲▼ ▲❖❸■ ❍❆■ ❱❰■ ◆❍❹◆
❍❊❘▼■❚■❆◆
✸✵
✸✳✶
✸✳✷
✸✳✸
✸✳✹

▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♥❤➙♥ ❍❡r♠✐t✐❛♥
❈→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♥❤➙♥ ❍❡r♠✐t✐❛♥
❈→❝ ❤➔♠ r✐➯♥❣ ❝õ❛ ♥❤➙♥ ❍❡r♠✐t✐❛♥ ✳
✣à♥❤ ❧þ ❍✐❧❜❡rt✲❙❝❤♠✐❞t ✳ ✳ ✳ ✳ ✳ ✳ ✳

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦


✷

✳
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✸✵
✸✷
✸✽
✹✺

✺✼
✺✽


ị

é

t ồ ữỡ tr ợ
ỡ ồ t ỵ ự ữợ t ỳ
ữỡ tr õ ữủ ồ ữỡ tr t Pữỡ tr t ữủ
ữ ởt ổ ử t ồ ỳ tr ỹ ữủ q t
ự t õ õ ự ử rở r ổ tr t ồ
ỏ tr ồ ử ữ ự ữỡ tr t
ợ qt ởt số t ỵ ữỡ tr
ổ t ổ t ữủ ữ tữủ t tữủ tr
ự ữỡ tr t õ trỏ q trồ tr ỵ tt t ồ
ợ ố ự t s ỡ ữỡ tr t ỗ

tớ õ õ t ởt số ớ t q tổ ỹ ồ
t Pữỡ tr t r tờ qt õ tốt
ồ

ệ

õ t tr ự s
ự ởt số ữỡ tr t õ t ữủ
ử ởt số ữỡ ữỡ tr t ởt số t
q

ìẹ

ự ởt số ữỡ tr t õ t ữủ

ệ

qt ỡ ừ t ữỡ tr t
õ t ữủ
ró ữỡ ữỡ tr t õ t ữủ

Pì PP

ữ t ồ ự t t tờ ủ tự
r ờ t ợ ữợ tr ụ ữ sr ợ tờ ở
ổ

ì P ế

ợ ừ õ

ởt ợ ố ợ t tr t ỗ tớ ụ ởt
ỏ ữ ữủ t ố ợ s P
ữợ t tr ừ õ
ự tờ ủ tố ữỡ tr t õ t ữủ

P ế

õ r ữủ ữỡ ởt số ữỡ tr t
t q

ể

ợ ử ữ õ ữủ t ữỡ ợ ỳ ở
s
ữỡ r ởt số tự ữỡ tr t ổ
ổ rt Pữỡ tr t t ũ
tr t q ữỡ s
ữỡ r ởt số ữỡ tr t r
ợ tờ qt
ữỡ r ởt số ữỡ tr t r
ợ rt



❈❤÷ì♥❣ ✶
❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❝➛♥ t❤✐➳t ❝❤♦ ❦❤â❛
❧✉➟♥ ♥❤÷✿ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✱ ✤➦❝ ❜✐➺t ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥
❧♦↕✐ ❤❛✐ ❋r❡❞❤♦❧♠ ✈î✐ ♥❤➙♥ t→❝❤ ❜✐➳♥✳


✶✳✶ ❑❤→✐ ♥✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ♠➔ ❤➔♠ ❝➛♥ t➻♠ ①✉➜t
❤✐➺♥ ❞÷î✐ ❞➜✉ t➼❝❤ ♣❤➙♥✳
❳➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝â ❞↕♥❣
b

λϕ(x) −

K(x, t)ϕ(t)dt = f (x),
a

tr♦♥❣ ✤â

✭✶✳✶✮

• f (x) ❧➔ ❤➔♠ ❝❤♦ tr÷î❝✱ ❝â ❣✐→ trà ♣❤ù❝ ✈➔ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a, b]❀
• K(x, t) ❧➔ ❤➔♠ ❝❤♦ tr÷î❝✱ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a, b] × [a, b]✱ ❝â ❣✐→ trà ♣❤ù❝ ✈➔ ✤÷ñ❝
❣å✐ ❧➔ ♥❤➙♥;
• λ ❧➔ ❤➡♥❣ sè ♣❤ù❝ ❝❤♦ tr÷î❝❀
• ϕ(x) ❧➔ ❤➔♠ ❝➛♥ t➻♠✱ ❧✉æ♥ ✤÷ñ❝ ❣✐↔ t❤✐➳t ❧➔ ❦❤↔ t➼❝❤ t❤❡♦ ♥❣❤➽❛ ❘✐❡♠❛♥♥✳ ❚❛ ❝â t❤➸

♣❤➙♥ ❧♦↕✐ ♥❤÷ s❛✉✿
✶✳ ◆➳✉ ❤➺ sè λ = 0 t❤➻ t❛ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤
b

K(x, t)ϕ(t)dt = f (x)
a

P❤÷ì♥❣ tr➻♥❤ tr➯♥ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❋r❡❞❤♦❧♠ ❧♦↕✐ ♠ët

✷✳ ◆➳✉ ❤➺ sè λ = 0 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠
❧♦↕✐ ❤❛✐✳
◆➳✉ ♥❤➙♥ K(x, t) ❝â t➼♥❤ ❝❤➜t K(x, t) ≡ 0 ✈î✐ ♠å✐ t > x t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ trð
t❤➔♥❤ ♣❤÷ì♥❣ tr➻♥❤ ❱♦❧t❡rr❛✳
✸✳ ◆➳✉ λ = 0 t❤➻ t❛ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤
x

λϕ(x) −

K(x, t)ϕ(t)dt = f (x),
a

✈➔ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❱♦❧t❡rr❛ ❧♦↕✐ ❤❛✐✳
✹


✹✳ ◆➳✉ λ = 0 t❤➻ t❛ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤
x

K(x, t)ϕ(t)dt = f (x),
a

✈➔ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❧♦↕✐ ♠ët✳
❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tæ✐ ❝❤➾ ①➨t ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐✳ ❇➡♥❣ ♣❤➨♣
❜✐➳♥ ✤ê✐✱ t❛ ❝â t❤➸ ✈✐➳t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ❞÷î✐ ❞↕♥❣
b

ϕ(x) = f (x) + λ

K(x, t)ϕ(t)dt.

a

✭✶✳✷✮

✶✳✷ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
❈❤ó♥❣ t❛ ❞ò♥❣ ❝→❝ ❦➼ ❤✐➺✉

Q [a, b] = [a, b] × [a, b] ,
C [a, b] = {f : [a, b] → C : ❢ ❧✐➯♥ tö❝ tr➯♥ [a, b]},
C(Q [a, b]}) = {f : Q [a, b]} → C : ❢ ❧✐➯♥ tö❝ tr➯♥ Q [a, b]},
R [a, b] ❧➔ t➟♣ ❤ñ♣ ❝→❝ ❤➔♠ ❣✐→ trà ♣❤ù❝ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ [a, b] ,
R2 [a, b] ❧➔ t➟♣ ❤ñ♣ ❝→❝ ❤➔♠ ❜➻♥❤ ♣❤÷ì♥❣ ❦❤↔ t➼❝❤ tr➯♥ [a, b] .
❱î✐ ♠é✐ f ∈ C [0, 1] , t❛ ❦➼ ❤✐➺✉
b

f

1

|f (x)|dx

=
a

✈➔

1/2

b


f

2

|f (x)|2 dx

=
a

❱î✐ ♠é✐ K(x) ∈ C(Q[0, 1]) t❛ ❦➼ ❤✐➺✉
b

K

2

1/2

b

|K(x, t)|2 dxdt

=
a

a

❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ t❤✉ë❝ C [a, b] t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛ t➼❝❤ ✈æ ❤÷î♥❣
b


f, g =

f (x)g(x)dx.
a

◆➳✉ f, g = 0 t❤➻ t❛ ♥â✐ f ✈➔ g trö❝ ❣✐❛♦✳
❚❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ Cauchy − Schwarz
b

b

a

b

|f (x)|2 dx

f (x)g(x)dx ≤
a

|g(x)|2 dx .
a

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ✭❍➺ ❝→❝ ❤➔♠ trü❝ ❝❤✉➞♥✮✳ ❉➣② {ϕn(x)} ❝→❝ ❤➔♠ t❤✉ë❝ C [a, b] ✤÷ñ❝
❣å✐ ❧➔ ♠ët ❤➺ trü❝ ❝❤✉➞♥ ♥➳✉

ϕn , ϕm =

0
1


✺

♥➳✉ n = m,
♥➳✉ n = m.


ừ = {n(x)}n=1 trỹ
f R2 [a, b] . f trỹ ợ ồ tỷ ừ r f
ữủ ồ ừ

t m = {1, ..., m} t ỳ ừ . f
ộ rr ở tử
r õ

span {m }

2

= 0 t

t t õ

f (x) = f, 1 1 (x) + ã ã ã + f, m m (x),

ữủ ồ số rr tự n ừ f (x)
ỹ ở tử {fn(x)} tr [a, b] .
õ {fn(x)} ở tử tợ f (x) tr [a, b] ợ ồ > 0, tỗ t số
N = N () s ợ ồ n N t | fn(x) f (x) |< ợ ồ x [a, b] .
f, n , n = 1, ..., m


ỵ ổ {fn(x)} tr [a, b]

ở tử ợ ồ > 0, tỗ t số N () s ợ ồ
n, m N () | fn (x) fm (x) |< ợ ồ x [a, b] .

ỵ {fn(x)}n=1 t ở tử tợ f (x) tr
[a, b] , t f (x) ụ t tr [a, b]
b

b

f (x)dx = lim

fn (x)dx.

n

a

a



ứ õ t s r r ộ un(x) ở tử S(x) tr [a, b] ợ ộ
n=1
n, un (x) t tr [a, b] t


b


b

S(x)dx =

un (x)dx.

a

a

n=1

ở tử tr
{fn (x)} tr R2 [a, b] ữủ ồ ở tử
2
tr tợ ợ f (x) tr R

[a, b]



1/2

b

lim fn f

n


2

2

|f (x) fn (x)| dx

= lim

n

= 0.

a

tỷ r K(x, t) tr Q [a, b]
t t tứ tr [a, b] t tỷ

K : R2 [a, b] R2 [a, b]
b

(t)

K(x, t)(t)dt
a

ồ t tỷ r tữỡ ự ợ t K(x, t) t
K1 (x, t) = K(x, t)
b

K2 (x, t) =

.............

K1 (x, s)K(s, t)ds
a
b

Km (x, t) =

Km1 (x, s)K(s, t)ds
a




❚❛ ❣å✐ Km(x, t) ❧➔ ♥❤➙♥ ❧➦♣ t❤ù m ❝õ❛ K(x, t)✳ ❚ø ✤à♥❤ ♥❣❤➽❛ t❛ ❝â
b

Km ϕ =

♠ = 1, 2....

Km (x, t)ϕ(t)dt,
a

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✽✳ ❚♦→♥ tû K ✤÷ñ❝ ❣å✐ ❧➔ ❜à ❝❤➦♥ ♥➳✉ tç♥ t↕✐ ❤➡♥❣ sè C ≥ 0 s❛♦ ❝❤♦
Kϕ

2

≤ C ϕ 2 , ∀ϕ ∈ R2 [a, b] .


◆➳✉ K ❜à ❝❤➦♥ t❤➻ t❛ ✤➦t
Kϕ 2
: ϕ ∈ R2 [a, b] , ϕ = 0
ϕ 2

K = sup

✈➔ ❣å✐ ♥â ❧➔ ❝❤✉➞♥ ❝õ❛ t♦→♥ tû K.

▼➺♥❤ ✤➲ ✶✳✷✳✾✳ ◆➳✉ ♥❤➙♥ K(x, t) ❝â
2
2

Kϕ
❈❤ù♥❣ ♠✐♥❤✳

≤ K

K(x, t)
2
2

ϕ 22 ,

2

< ∞ t❤➻

∀ϕ ∈ R2 [a, b]


⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ Cauchy − Schwarz t❛ ❝â
2

b
2

|Kϕ(x)|

=

K(s, x)ϕ(s)ds
a
b

b

|K(x, s)|2 ds

≤

|ϕ(s)|2 ds .

a

a

❉♦ ✈➟②
b


Kϕ

2
2

|Kϕ(x)|2 dx

=
a

b

2

≤

|ϕ(s)|2 ds

|K(x, s)| dsdx
a

=

b

b

K

a

2
2

ϕ

a
2
2.

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✵✳ ●✐↔ sû K ❧➔ t♦→♥ tû ❋r❡❞❤♦❧♠ t÷ì♥❣ ù♥❣ ✈î✐ ♥❤➙♥ K(x, t). ❑➼ ❤✐➺✉

K ∗ (x.t) = K(x, t)

✈➔ t♦→♥ tû K∗ ①→❝ ✤à♥❤ ❜ð✐

b
∗

2

∗

K ∗ (x, t)ϕ(t)dt

K : ϕ ∈ R [a, b] → (K ϕ) (x) =
a

✤÷ñ❝ ❣å✐ ❧➔ t♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ t♦→♥ tû K.
❚ø ✤à♥❤ ♥❣❤➽❛ t❛ s✉② r❛✿
✭✐✮ ❱î✐ ♠å✐ ϕ, ψ t❤✉ë❝ R2 [a, b] t❤➻ Kϕ, ψ

✭✐✐✮ ❱î✐ ♠å✐ m ≥ 1 t❤➻ (Km)∗ = (K∗)m.
✼

= ϕ, K∗ ψ .


✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✶✳ ✭▼✐➲♥✮✳ ❚➟♣ Ω ⊂ C ♠ð✱ ❦❤→❝ ré♥❣ ✈➔ ❧✐➯♥ t❤æ♥❣ ✤÷ñ❝ ❣å✐ ❧➔ ♠ët

♠✐➲♥ tr♦♥❣ C.

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✷✳ ❈❤♦ ♠✐➲♥ Ω ✈➔ ❤➔♠ f : Ω → C.

✭✐✮ ❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ tr➯♥ Ω ♥➳✉ f ❦❤↔ ✈✐ t↕✐ ♠å✐ ✤✐➸♠ z ∈ Ω.
✭✐✐✮ ❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ Ω ♥➳✉ tç♥ t↕✐ t➟♣ P ⊂ Ω s❛♦ ❝❤♦✿
✲ P ❦❤æ♥❣ ❝â ✤✐➸♠ ❣✐î✐ ❤↕♥ tr♦♥❣ Ω;
✲ f (z) ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ tr♦♥❣ ♠✐➲♥ Ω\P ;
✲ ▼å✐ ✤✐➸♠ ❝õ❛ P ✤➲✉ ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ f (z).
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✸✳ ❍➔♠ f : C → C ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ♥❣✉②➯♥ ♥➳✉ f ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤
tr➯♥ t♦➔♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C.

✶✳✸ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ✈î✐
♥❤➙♥ t→❝❤ ❜✐➳♥
❚r♦♥❣ ♠ö❝ ♥➔②✱ t❛ ①➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ❝â ❞↕♥❣
b

ϕ(x) = f (x) + λ

K(x, t)ϕ(t)dt,
a


✭✶✳✸✮

tr♦♥❣ ✤â K(x, t) ❧➔ ♥❤➙♥ t→❝❤ ❜✐➳♥ tr➯♥ Q [a, b] ❝â ❞↕♥❣
n

K(x, t) =

ai (x)bi (t),

✭✶✳✹✮

i=1

tr♦♥❣ ✤â ai(x), bi(t) ❧➔ ❝→❝ ❤➔♠ t❤✉ë❝ C [a, b] . ❚❤❛② ✭✶✳✹✮ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ t❛ t❤✉
✤÷ñ❝
n

ϕ(x) = f (x) + λ

b

ai (x)

bi (t)ϕ(t)dt .
a

i=1

❑❤✐ ✤â ♣❤÷ì♥❣ tr✐♥❤ tr➯♥ trð t❤➔♥❤
n


ci ai (x),

ϕ(x) = f (x) + λ

✭✶✳✺✮

i=1

tr♦♥❣ ✤â ci = ab bi(t)ϕ(t)dt, i = 1, ..., n. ❚ø ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ s✉② r❛ ♥❣❤✐➺♠ ϕ(x) ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ ✤÷ñ❝ ①→❝ ✤à♥❤ ♥➳✉ ♥❤÷ ①→❝ ✤à♥❤ ✤÷ñ❝ ❤➺ sè ci.
◆❤➙♥ ❝↔ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ ✈î✐ bi(t) rç✐ ❧➜② t➼❝❤ ♣❤➙♥ t❤❡♦ ❜✐➳♥ t tr➯♥ [a, b]
t❛ t❤✉ ✤÷ñ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤
n

aij cj ,

ci = f i + λ

i = 1, ..., n,

j=1

tr♦♥❣
b

fi =

b


bi (t)f (t)dt,

aij =

a

aj (t)bi (t)dt.
a

✽


❱✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t÷ì♥❣ ✤÷ì♥❣ ✈î✐ ✈✐➺❝ ❣✐↔✐ ♠ët ❤➺ ✤↕✐ sè t✉②➳♥ t➼♥❤
(■ − λ❆)❝ = ❢,

✭✶✳✻✮
tr♦♥❣ ✤â ■ ❧➔ ♠❛ tr➟♥ ✤♦♥ ✈à ❝➜♣ n × n, ❆ = (aij ) ❧➔ ♠❛ tr➟♥ ❝➜♣ n × n ✈î✐ ❝→❝ ♣❤➛♥ tû
✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ tr➯♥✱ ❢ = (f1, ..., fn)T ✈➔ ❝ = (c1, ..., cn)T ❧➔ ❝→❝ ❤➺ sè ♣❤↔✐ t➻♠✳
❑➼ ❤✐➺✉ D(λ) = ❞❡t (■ − λ❆) . ❱✐➺❝ ❣✐↔✐ ❤➺ t✉②➳♥ t➼♥❤ ✭✶✳✻✮ ♣❤ö t❤✉ë❝ ✈➔♦ ❣✐→ trà ❝õ❛
❚❛ ①➨t ❤❛✐ tr÷í♥❣ ❤ñ♣ s❛✉✿
❚r÷í♥❣ ❤ñ♣ ✶✿ D(λ) = 0. ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② λ ✤÷ñ❝ ❣å✐ ❧➔ ❣✐→ trà ❝❤➼♥❤ q✉② ❝õ❛
♥❤➙♥✳ ❑❤✐ ✤â ❤➺ t✉②➳♥ t➼♥❤ tr➯♥ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t

D(λ).

❝ = (■ − λ❆)−1❢,

❤❛②

1

❝ = D(λ)
adj(■ − λ❆❢,

tr♦♥❣ ✤â ❛❞❥(■ − λ❆) = (Dji(λ))❧➔ ♠❛ tr➟♥ ♣❤ö ❤ñ♣ ❝õ❛ ♠❛ tr➟♥ (■ − λ❆). ❉♦ ✤â ♠é✐
❤➺ sè ci ❝â ❜✐➸✉ ❞✐➵♥
1
ci =
D(λ)

n

Dji (λ)fj .
j=1

❚❤❛② ❜✐➸✉ ❞✐➵♥ ❝õ❛ ci ✈➔ fi ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ t❛ ✤÷ñ❝
b

ϕ(x) = f (x) + λ
a

1
D(λ)

n

Dji (λ)ai (x)bj (t) f (x)dt.
i=1 j=1

❑➼ ❤✐➺✉
1

R(x, t; λ) =
D(λ)

n

n

n

Dji (λ)ai (x)bj (t)
i=1 j=1

✈➔ ❣å✐ ♥â ❧➔ ♥❤➙♥ ❣✐↔✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳ ❑❤✐ ✤â ♥❣❤✐➺♠ ϕ(x) ✤÷ñ❝ ①→❝ ✤à♥❤
❜ð✐ ❝æ♥❣ t❤ù❝
b

ϕ(x) = f (x) + λ

✭✶✳✼✮

R(x, t; λ)f (t)dt.
a

✣➦t



❉(x, t; λ) = 





0
a1 (x)
a2 (x) · · ·
b1 (t) 1 − λa11 −λa12 · · ·
b2 (t) −λa21 1 − λa22 · · ·

✳✳✳

✳✳✳

✳✳✳

bn (t)

−λan1

−λan2

✈➔ D(x, t; λ) = ❞❡t(❉(x, t; λ)). ❑❤✐ ✤â

R(x, t; λ) = −

✾

D(x, t; λ)
.
D(λ)


✳✳✳

an (x)
−λa1n
−λa2n

✳✳✳

· · · 1 − λann









✭✶✳✽✮


❚r÷í♥❣ ❤ñ♣ ✷✿ D(λ) = 0. ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② λ ✤÷ñ❝ ❣å✐ ❧➔ ❣✐→ trà r✐➯♥❣ ❝õ❛
♥❤➙♥✳ ●✐↔ sû λk ❧➔ ♠ët ❣✐→ trà r✐➯♥❣✱ ♥❣❤➽❛ ❧➔ D(λk ) = 0.
❳➨t tr÷í♥❣ ❤ñ♣ f = 0. ❑❤✐ ✤â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ trð t❤➔♥❤
(■ − λk ❆)❝ = 0.

❱➻ D(λk ) = 0 ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❝â pk ♥❣❤✐➺♠ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✤÷ñ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐


❝(j)(λk ) = 



(j)



c1 (λk )
(j)

✳✳✳

j = 1, ..., pk .




cn (λk )

❚❤❛② ❝→❝ ❣✐→ trà ♥➔② ✈➔♦ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ t❛ t❤✉ ✤÷ñ❝ ❝→❝ ♥❣❤✐➺♠
n
(j)

ϕj (x; λk ) = f (x) + λk

ci (λk )a)i (x),

j = 1, ..., pk

i=1


◆➳✉ f (x) ≡ 0 tr➯♥ [a, b] t❤➻ ♠é✐ ❤➔♠
n
(e)
ϕj (x; λk )

(j)

= λk

ci (λk )a)i (x)
i=1

❧➔ ♠ët ♥❣❤✐➺♠ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t❤✉➛♥ ♥❤➜t
b

K(x, t)ϕ(t)dt.

ϕ(x) = λk
a

❈❤➾ sè tr➯♥ (e) ✤➸ ❦➼ ❤✐➺✉ r➡♥❣ ♥❣❤✐➺♠ ϕ(e)
j (x; λ) ❧➔ ❤➔♠ r✐➯♥❣ ❝õ❛ ♥❤➙♥ t÷ì♥❣ ù♥❣ ✈î✐
❣✐→ trà r✐➯♥❣ λk . ▼é✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t s➩ ❝â ❞↕♥❣
pk

ϕ

(h)

(e)


(x; λk ) =

αj ϕj (x; λk ),
j=1

tr♦♥❣ ✤â αj ❧➔ ❤➡♥❣ sè tò② þ✱ ❝❤➾ sè tr➯♥ (h) ❧➔ ✤➸ ❦➼ ❤✐➺✉ r➡♥❣ ϕ(h)(x; λk ) ❧➔ ♥❣❤✐➺♠ tê♥❣
q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t❤✉➛♥ ♥❤➜t tr➯♥✳
❳➨t tr÷í♥❣ ❤ñ♣ ❢ = 0. ❚❛ s➩ sû ❞ö♥❣ ❜ê ✤➲ s❛✉✿

❇ê ✤➲ ✶✳✸✳✶✳ ❈❤♦ B = (bij )n×n ✈➔ B∗ = (bij )n×n ✳ ❑❤✐ ✤â ♥➳✉ ❞❡t (B) = 0 t❤➻ ❤➺

❦❤æ♥❣ t❤✉➛♥ ♥❤➜t Bx = f ❝â ♥❣❤✐➺♠ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f trü❝ ❣✐❛♦ ✈î✐ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠
❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧✐➯♥ ❤ñ♣ t❤✉➛♥ ♥❤➜t B∗ y = 0.

❚ø ❜ê ✤➲ ♥➔②✱ t❛ t❤➜② ❤➺ t✉②➳♥ t➼♥❤ (I − λk A)c = f ❝â ♥❣❤✐➺♠ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f trü❝
❣✐❛♦ ✈î✐ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
(I − λk A)∗ d = 0.
✭✶✳✾✮
❱➻ ♠❛ tr➟♥ (I − λk A) ✈➔ (I − λk A)∗ ❝â ❝ò♥❣ ❤↕♥❣ ✈➔ sè ❦❤✉②➳t ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✾✮
❝ô♥❣ ❝â pk ♥❣❤✐➺♠ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ▲↕✐ ❝â
❚

(I − λk A)∗ d = (I − λk A∗ )d = (I − λk A )d = 0.

✶✵


❱➻ ✈➟② ♥➳✉


❚
d(j) (λk ) = (d1 (λk ), ..., d(j)
n (λk ))
(j)

❧➔ ♠ët tr♦♥❣ pk ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✾✮ t❤➻
n

d(j)
m (λk )

(j)

− λk

aim di (λk ) = 0,

m = 1, ..., n.

✭✶✳✶✵✮

i=1

▼➦t ❦❤→❝✱ ①➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t❤✉➛♥ ♥❤➜t ❧✐➯♥ ❤ñ♣ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮
b

ψ(x) = λ

K(t, x)ψ(t)dt.
a


✭✶✳✶✶✮

n

❱➻ K(t, x) = ai(t)bi(x) ♥➯♥ t÷ì♥❣ tü ♥❤÷ tr➯♥✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮ ❝â t❤➸ ✤÷ñ❝ ✈✐➳t
n=1
❞÷î✐ ❞↕♥❣
n
dm − λ
aim di = 0,
m = 1, ..., n
✭✶✳✶✷✮
i=1

tr♦♥❣ ✤â di = ab ai(t)ψ(t)dt✱ aim = ab am(t)bi(t)dt. ❚ø ✭✶✳✶✵✮ ✈➔ ✭✶✳✶✷✮ s✉② r❛ d(j)(λk )
❧➔ ♥❣✐➺♠ ❝õ❛ ✭✶✳✸✮ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✶✷✮ ✈î✐ λ = λk .
❚ø ✤â t❛ ❝â ❦➳t q✉↔ s❛✉✿

✣à♥❤ ❧þ ✶✳✸✳✷✳ ✭✣à♥❤ ❧þ ❋r❡❞❤♦❧♠ ✤è✐ ✈î✐ ❤↕t ♥❤➙♥ t→❝❤ ❜✐➳♥✮✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤
♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐

b

K(x, t)ϕ(t)dt

ϕ(x) = f (x) + λ
a

tr♦♥❣ ✤â λ ❧➔ t❤❛♠ sè ♣❤ù❝✱ f (x) ∈ C[a, b] ✈➔ K(x, t) ∈ C(Q[a, b]) ❧➔ ❤↕t ♥❤➙♥ t→❝❤ ❜✐➳♥

❝â ❞↕♥❣ ✭✶✳✹✮✳ ❑❤✐ ✤â✿
✭✐✮ ♥➳✉ λ ❧➔ ❣✐→ trà ❝❤➼♥❤ q✉② ❝õ❛ ♥❤➙♥ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❜✐➸✉ ❞✐➵♥
❞÷î✐ ❞↕♥❣
b

ϕ(x) = f (x) + λ

R(x, t; λ)f (t)dt,
a

tr♦♥❣ ✤â R(x, t; λ) ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ ✭✶✳✽✮
✭✐✐✮ ◆➳✉ λ ❧➔ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♥❤➙♥ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t
b

K(x, t)ϕ(t)dt

ϕ(x) = λ
a

❝â ♥❣❤✐➺♠ ❦❤æ♥❣ t➛♠ t❤÷í♥❣✳ ❍ì♥ ♥ú❛✱ ♣❤÷ì♥❣ tr➻♥❤ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ❝â ♥❣❤✐➺♠ ❦❤✐ ✈➔
❝❤➾ ❦❤✐ ❤➔♠ f (x) trü❝ ❣✐❛♦ ✈î✐ t➜t ❝↔ ❝→❝ ❤➔♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❧✐➯♥ ❦➳t
b

ψ(x) = λ

K(t, x)ψ(t)dt.
a

✶✶



❈❤÷ì♥❣ ✷
P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆
❋❘❊❉❍❖▲▼ ▲❖❸■ ❍❆■ ❱❰■
◆❍❹◆ ❚✃◆● ◗❯⑩❚
Ð ❈❤÷ì♥❣ ✶✱ t❛ ✤➣ ①➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ✈î✐ K(x, t) ❧➔ ♥❤➙♥
t→❝❤ ❜✐➳♥ t❤✉ë❝ C(Q[a, b]). ❚❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧þ ❋r❡❞❤♦❧♠ ✈➲ ❝➜✉ tró❝ ❝õ❛ ❝→❝
♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♣❤ö t❤✉ë❝ ✈➔♦ λ. ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣
tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ✈î✐ ♥❤➙♥ tê♥❣ q✉→t✳

✷✳✶ P❤÷ì♥❣ ♣❤→♣ t❤➳ ❧✐➯♥ ❤ñ♣

✣à♥❤ ❧þ ✷✳✶✳✶✳ ✭✣à♥❤ ❧þ t❤❛② t❤➳ ❧✐➯♥ t✐➳♣✮✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐
❤❛✐

b

ϕ(x) = f (x) + λ

K(x, t)ϕ(t)dt.
a

✭✷✳✶✮

tr♦♥❣ ✤â λ ❧➔ ♠ët t❤❛♠ sè ♣❤ù❝✱ f (x) ∈ C[a, b] ✈➔ K(x, t) ❧➔ ♠ët ♥❤➙♥ t❤✉ë❝ C(Q[a, b]).
◆➳✉ |λ|(b − a) sup |K(x, t)| < 1 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ✤÷ñ❝ ①→❝
(x,t)∈Q[a,b]

✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝
b


ϕ(x) = f (x) + λ

R(x, t; λ)f (t)dt.
a

tr♦♥❣ ✤â R(x, t; λ) ❧➔ t♦→♥ tû ❣✐↔✐ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐
∞

λm−1 Km (x, t).

R(x, t; λ) =
m=1

✣➦t M = sup |K(x, t)|. ●➾❛ sû ϕ(x) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
(x,t)∈Q[a,b]
✭✷✳✶✮✳ ❑❤✐ ✤â t❤❛② ❜✐➸✉ t❤ù❝ ❝õ❛ ϕ(x) ✈➔♦ ϕ(t) tr♦♥❣ ✈➳ ♣❤↔✐ t❛ t❤✉ ✤÷ñ❝
❈❤ù♥❣ ♠✐♥❤✳

b

ϕ(x) = f (x) + λ

b

K(x, t) f (t) + λ
a

K(t, s)ϕ(s)ds dt
a


b

b

b

K(x, t)f (t)dt + λ2

= f (x) + λ
a

K(x, t)K(t, s)ϕ(s)dsdt.
a

✶✷

a


❙❛✉ ❦❤✐ t❤❛② ✤ê✐ t❤ù tü ❧➜② t➼❝❤ ♣❤➙♥ tr♦♥❣ t➼❝❤ ♣❤➙♥ ❝✉è✐ ✈➔ t❤❛② t❤➳ ❜✐➳♥ s ✈î✐ t✱ t❛
t❤✉ ✤÷ñ❝
b

b
2

ϕ(x) = f (x) + λ

K(x, t)f (t)dt + λ

a

K2 (x, t)ϕ(t)dt.
a

❚✐➳♣ tö❝ q✉→ tr➻♥❤ ♥➔② n ❧➛♥✱ t❛ s➩ t❤✉ ✤÷ñ❝ ❞↕♥❣ tê♥❣ q✉→t
n

b

λm

ϕ(x) = f (x) +

b

Km (x, t)f (t)dt

+ λn+1

Kn+1 (x, t)ϕ(t)dt.

a

m=1

a

✭✷✳✷✮


✣➦t
n

b

σn (x) =

λ

m−1

Km (x, t)f (t)dt ,
a

m=1
b

ρn (x) = λn+1

Kn+1 (x, t)ϕ(t)dt.
a

❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ❝â t❤➸ ✈✐➳t ❣å♥ ❧↕✐ ❧➔
ϕ(x) = f (x) + λσn (x) + ρn (x).

❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ |λ|M (b − a) < 1 t❤➻ ❞➣② {σn(x)} ❤ë✐ tö ✤➲✉ ✤➳♥ ❤➔♠ ❧✐➯♥
tö❝ σ(x) tr➯♥ ✤♦↕♥ [a, b] ✈➔ ❞➣② {ρn(x)} ❤ë✐ tö ✤➲✉ ✤➳♥ ✵ tr➯♥ ✤♦↕♥ [a, b]. ❚❤➟② ✈➟②✱ ✈➻
|K(x, t)| ≤ M ♥➯♥ |Km (x, t)| ≤ M m (b − a)m−1 . ❉♦ ✤â
b


λm−1

Km (x, t)f (t)dt ≤ (|λ|M (b − a))m−1 M f

1.

a

❱➻ |λ|M (b − a) < 1 ♥➯♥ ❦❤✐ n ✤õ ❧î♥ t❤➻ ∀m > n, ∀ε > 0, t❛ ❝â
m

(|λ|M (b − a))k−1 M f

|σm (x) − σn (x)| ≤

1

k=n+1

≤ (|λ|M (b − a))n

M f 1
1 − |λ|M (b − a)

< ε.

◆❤÷ ✈➟②✱ ❞➣② ❤➔♠ ❧✐➯♥ tö❝ σn(x) ❤ë✐ tö t✉②➺t ✤è✐ ✈➔ ✤➲✉ tî✐ ❤➔♠ ❣✐î✐ ❤↕♥
∞

b


σ(x) =

λ
m=1

m−1

Km (x, t)f (t)dt .
a

❍ì♥ ♥ú❛✱ ❧➜② t➼❝❤ ♣❤➙♥ tø♥❣ sè ❤↕♥❣ t❛ ✤÷ñ❝
b

∞

σ(x) =

b

λ
a

m=1

m−1

▼➦t ❦❤→❝✱ t❛ ❝â |ρn(x)| ≤ |λ|M
[a, b] ❦❤✐ n → +∞.


Km (x, t) f (t)dt =

R(x, t; λ)f (t)dt.
a

ϕ 1 (|λ|M (b − a))n . ❉♦ ✤â ρn (x) ❤ë✐ tö ✤➲✉ tî✐ ✵ tr➯♥

✶✸


❱➟② ϕ(x) = f (x) + λσ(x) ❤❛②
b

ϕ(x) = f (x) + λ

R(x, t; λ)f (t)dt.
a

❱➼ ❞ö ✷✳✶✳✷✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ s❛✉✿
ϕ(x) = cos x +

π/2

1
2

sin xϕ(t)dt.
0

▲í✐ ❣✐↔✐✳ ❚r÷î❝ ❤➳t t❛ t❤➜②

sup | sin x| = 1,
[0, π2 ]

1
λ= ,
2

a = 0,

b=

π
.
2

❙✉② r❛
1 π
π
λ(b − a) sup |K(x, t)| = ( − 0)1 = < 1.
2 2
4

❱➻ t❤➳ t❛ ❝â t❤➸ →♣ ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ t❤❛② t❤➳ ❧✐➯♥ t✐➳♣ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ♥➔②✳ ❚❤❛②
ϕ(t) = cos t +

1
2

✈➔♦ ❜✐➸✉ t❤ù❝ ❞÷î✐ ❞➜✉ t➼❝❤ ♣❤➙♥✱ t❛ ✤÷ñ❝
ϕ(x) = cos x +


1
2

π/2

sin tϕ(t1 )dt1
0

π/2

sin x cos t +
0

1
2

π/2

sin tϕ(t1 )dt1 dt
0

1 π/2
1 π/2 π/2
= cos x +
sin x cos tdt +
sin x sin tϕ(t1 )dt1 dt
2 0
4 0
0

π/2
1
1 π/2
= cos x + sin x +
sin x
sin tϕ(t1 )dt1 dt.
2
4 0
0

▲↕✐ t❤❛② ϕ(t1) = cos t1 + 21
1
sin x +
2
1
= cos x + sin x +
2

ϕ(x) = cos x +

π/2
sin t1 ϕ(t2 )dt2 ,
0

t❛ ✤÷ñ❝

1 π/2
1
sin x sin t cos t1 dt1 dt +
2 0

4
π/2
1
1
sin x + sin x
ϕ(t2 )dt2 .
4
8
0

❈ù t✐➳♣ tö❝ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② t❛ s➩ t❤✉ ✤÷ñ❝ ♥❣❤✐➺♠
ϕ(x) = cos t + sin x

1 1 1
+ + + ···
2 4 8

= cos x + sin x.

✶✹

π/2

π/2

sin x sin t sin t1 ϕ(t2 )d
0

0



Pữỡ t

r t s ợ t ởt ữỡ qt ữỡ tr
t r t ủ ừ ữỡ t s sỷ ử
ự sỹ ở tử t ữủ ởt t q tốt ỡ tr trữớ ủ ộ t
tỷ õ ở tử ợ
t ữỡ tr t r
b

(x) = f (x) +

K(x, t)(t)dt.
a



t t ồ ổ 0(x) ởt tr tỹ ợ a x b. ổ
tữớ t s ồ 0(x) {0, 1, x, ex} . ởt 1(x) ừ (x) ữủ
ữ s
b

1 (x) = f (x) +

K(x, t)0 (t)dt.
a



2(x) ừ (x) t ữủ t t (t) tr ữỡ

tr 1(x) t ữủ
b

2 (x) = f (x) +

K(x, t)1 (t)dt
a

ự t tử ữỡ t s t ữủ n + 1 ừ (x) t
ổ tự tr ỗ s
b

K(x, t)n (t)dt,

n+1 (x) = f (x) +

ợ n 0.

a

t ộ j (x) tr tú ừ t t
j = 0, ..., n t t ữủ
n

b
m



n+1 (x) = f (x) +


Km (x, t)f (t)dt
a

m=1
b

+

n+1

Kn+1 (x, t)0 (t)dt
a


n+1 (x) = f (x) + n (x) + n+1 (x),

tr õ
n

n (x) =

b



m1

Km (x, t)f (t)dt ,
a


m=1
b

n+1 (x) =

ứ õ ỵ s

n+1

Kn+1 (x, t)0 (t)dt.
a



j+1 (x)

ợ


✣à♥❤ ❧þ ✷✳✷✳✶✳ ✭✣à♥❤ ❧þ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣ ✮✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐✳
b

ϕ(x) = f (x) + λ

✭✷✳✺✮

K(x, t)ϕ(t)dt,
a


tr♦♥❣ ✤â λ ❧➔ ♠ët t❤❛♠ sè ♣❤ù❝✱ f (x) ∈ C[a, b] ✈➔ ❝❤♦ K(x, t) ❧➔ ♠ët ♥❤➙♥ t❤✉ë❝
C(Q[a, b]). ◆➳✉ |λ| K 2 < 1 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ✈➔ ♥❣❤✐➺♠ ✤â
✤÷ñ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝
b

ϕ(x) = f (x) + λ

R(x, t; λ)f (t)dt,
a

tr♦♥❣ ✤â R(x, t; λ) ❧➔ t♦→♥ tû ❣✐↔✐ ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉
∞

λm−1 Km (x, t).

R(x, t; λ) =
m=1

❚❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ♠➔ ✤➣ t❤ü❝ ❤✐➺♥ ð tr➯♥✱ t❛ ✤➣ ❝â ①➜♣ ①➾ ❜➟❝ n + 1
❝õ❛ ♥❣❤✐➺♠ ϕ(x) ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝
❈❤ù♥❣ ♠✐♥❤✳

ϕn+1 (x) = f (x) + λσn (x) + ωn+1 (x).

❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ |λ| K 2 < 1 t❤➻ ❞➣② σn(x) ❤ë✐ tö ✤➲✉ tî✐ ❤➔♠ ❣✐î✐ ❤↕♥ σ(x),
❝á♥ ❞➣② ωn+1(x) ❤ë✐ tö ✈➲ ✵ ❦❤✐ n → ∞. ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✤è✐
✈î✐ ❤↕t ♥❤➙♥ ❧➦♣ t❛ ✤÷ñ❝
b
2


b
2

|Km (x, t)| ≤

|K(s, t)|2 ds .

|Km−1 (x, s)| ds
a

a

▲➜② t➼❝❤ ♣❤➙♥ tr➯♥ ❝↔ ❤❛✐ ✈➳ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② t❤❡♦ ❜✐➳♥ t✱ t❛ ✤÷ñ❝
b

b
2

|Km (x, t)| dt ≤
a

b

b

2

|K(s, t)|2 dsdt .

|Km−1 (x, s)| ds

a

a

a

✣➦t
b

|Km (x, t)|2 dt.

κm (x) =
a

❚❛ ✤÷ñ❝

κm (x) ≤ κm−1 (x) K

❇➡♥❣ q✉② ♥↕♣✱ t❛ t❤✉ ✤÷ñ❝ ✤→♥❤ ❣✐→

κm (x) ≤ κ1 (x) K

2
2

.

2m−2
.
2


⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✤è✐ ✈î✐ ♥❤ú♥❣ t➼❝❤ ♣❤➙♥ tr♦♥❣ tê♥❣ σn(x),
t❛ ✤÷ñ❝

✶✻


2

b

b

≤

Km (x, t)f (t)dt

b
2

|f (t)|2 dt

|Km (x, t)| dt
a

a

a
2
2

2
2

≤ κm (x) f
≤ κ1 (x) f

K

2m−2
.
2

❉♦ ✤â ♠é✐ sè ❤↕♥❣ tr♦♥❣ tê♥❣ σn(x) ✤➲✉ ✤÷ñ❝ ✤→♥❤ ❣✐→ ❜ð✐ ❜➜t ✤➥♥❣ t❤ù❝
b

λ

m

κ1 (x) f
K 2

Km (x, t)f (t)dt ≤
a

2

(|λ| K 2 )m .

❚ø ✤â t❛ s✉② r❛ r➡♥❣ ♥➳✉ ♥❤÷ |λ| K 2 < 1 t❤➻ ❝❤✉é✐ {σn(x)} ❤ë✐ tö t✉②➺t ✤è✐ ✈➔ ✤➲✉ tî✐

❤➔♠ ❣✐î✐ ❤↕♥ ❞✉② ♥❤➜t σ(x) tr➯♥ ✤♦↕♥ [a, b]. ❈ô♥❣ sû ❞ö♥❣ ✤→♥❤ ❣✐→ t÷ì♥❣ tü ♥❤÷ tr➯♥✱
t❛ t❤➜② r➡♥❣
κ1 (x) ϕ0
K 2

|ωn+1 (x)| ≤

2

(|λ| K 2 )n+1 → 0 khi n → +∞.

◆❤÷ ✈➟② tø ❤❛✐ ✤✐➲✉ tr➯♥ t❛ s✉② r❛ r➡♥❣
ϕ(x) = f (x) + λσ(x).
❈❤ù♥❣ ♠✐♥❤ t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ✿ ●➾❛ sû ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✮ ❝â ❤❛✐ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t

✈➔ ϕ(x). ✣➦t δ(x) = ϕ(x) − ϕ(x). ❑❤✐ ✤â δ(x) t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥
t❤✉➛♥ ♥❤➜t
ϕ(x)

b

δ(x) = λ

K(x, t)δ(t)dt.
a

⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③ t❛ ✤÷ñ❝
b
2


|δ(x)| ≤ |λ|

2

b
2

|δ(t)|2 dt .

|K(x, t)| dt
a

a

▲➜② t➼❝❤ ♣❤➙♥ ❝↔ ❤❛✐ ✈➳ t❤❡♦ ❜✐➳♥ x s✉② r❛
b
2

(1 − |λ| K

2
2)

|δ(x)|2 dx ≤ 0.
a

◆❤÷♥❣ ✈➻ |λ| K 2 < 1 ❝❤♦ ♥➯♥ ab |δ(x)|2dx = 0, s✉② r❛ δ(x) = 0 ❤❛② ϕ(x) ≡ ϕ(x).
❈❤ù♥❣ tä ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t✳
❱➼ ❞ö ✷✳✷✳✷✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ s❛✉
1

ϕ(x) = cosx +
2

▲í✐ ❣✐↔✐✳ ❚❛ ❦✐➸♠ tr❛ t❤➜② r➡♥❣
K(x, t)

1

cos(xt)ϕ(t)dt.
0

2

✶✼

1
=√ .
2


õ
|| K(x, t)

=

2

1 1
< 1.
2 2


t õ t ử ữỡ t ữỡ tr t
t t ổ 0(x) = 1. õ ởt
1 1
1 (x) = cosx +
cost0 (t)dt.
2 0
1 sinx
.
= cosx +
2 x

ự t tử ữỡ t t ữủ ừ ữỡ tr t tr
1 (x) =

11532090
7944195 2
607005 4

x +
x.
6397711
12795422
12795422

ỵ r
é ữỡ t ự ỵ r ố ợ ữỡ tr t
r
b


K(x, t)(t)dt,

(x) = f (x) +
a



ợ t é ử t r ữỡ t ữỡ
tr ữ K(x, t) tờ qt ợ || < K1 . é t s t ủ
ữỡ ỹ ỵ r ố ợ K(x, t) C(Q[a, b])
tờ qt t số ự tũ ỵ
rữợ t t sỷ t số ự tở õ = { : || } , tr õ
số ố õ t ợ tũ ỵ s t t K(x, t).
ỵ rstrss K(x, t) õ t t t tờ ừ
tử tr ự
2



K(x, t) = Ksep (x, t) + K (x, t),

tr õ Ksep(x, t) tự t õ
n

Ksep (x, t) =

ai (x)bi (t),
i=1

tr õ ai(x) bi(t) tở C[a, b]. ỏ K(x, t) tọ

b

K

2

1/2

b
2

|K (x, t)| dxdt

=
a

< .

a

ồ = 1 õ || t || <
ữỡ tr ữợ

1
K



2


.

ứ sỹ t t õ t t


b

ϕ(x) = f (x) + λ

b

Ksep (x, t)ϕ(t)dt + λ
a

✣➦t F (x; λ) = f (x) + λ

Kε (x, t)ϕ(t)dt.
a

t❤➻

b
a Ksep (x, t)ϕ(t)dt

b

ϕ(x) = F (x; λ) + λ

Kε (x, t)ϕ(t)dt.
a


✭✷✳✽✮

❱➻ ϕ(t) ❦❤↔ t➼❝❤ ♥➯♥ F (x; t) ❧✐➯♥ tö❝ tr➯♥ [a, b]. ❱➻ λ < K1 ♥➯♥ t❤❡♦ ✤à♥❤ ❧þ ①➜♣ ①➾ ❧✐➯♥
t✐➳♣✱ ♥❣❤✐➺♠ ϕ(x) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✽✮ tç♥ t↕✐✱ ❧✐➯♥ tö❝ tr➯♥ [a, b] ✈➔ ✤÷ñ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐
ε 2

b

ϕ(x) = F (x; λ) + λ

Rε (x, t; λ)F (t; λ)dt,
a

tr♦♥❣ ✤â

∞

✭✷✳✾✮

λm−1 Kεm (x, t)

Rε (x, t; λ) =
m=1

✈➔ Kεm ❧➔ ♥❤➙♥ ❧➦♣ ❝õ❛ Kε(x, t). ❚❤❛② F (x; λ) = f (x) + λ
♣❤↔✐ t❛ ✤÷ñ❝

b
a Ksep (x, t)ϕ(t)dt


b

ϕ(x) = fε (x; λ) + λ

Gε (x, t; λ)ϕ(t)dt,

✭✷✳✶✵✮

Rε (x, t; λ)f (t)dt,

✭✷✳✶✶✮

a

tr♦♥❣ ✤â

b

fε (x; λ) = f (x) + λ
a

✈➔

b

Gε (x, t; λ) = Ksep (x, t) + λ

Rε (x, u; λ)Ksep (u, t)du.
a


❚❛ t❤➜② r➡♥❣ G(x, t; λ). ❝â ❞↕♥❣ t→❝❤ ❜✐➳♥ ❜ð✐ ✈➻
Ksep (x, t) = ai (x)bi (t),

✈➔
b

n

b

Rε (x, u; λ)Ksep (u, t)du =
a

Rε (x, u; λ)
a

ai (u)bi (t) du
i=1

n

b

=

Rε (x, u; λ)ai (u)du bi (t)
i=1
n


=

a

Aεi (x; λ)bi (t),
i=1

✶✾

✈➔♦ ✈➳

✭✷✳✶✷✮


tr õ
b

Ai (x; ) =

R (x, u; )ai (u)du
a


b

m1 Km (x, u) ai (u)du

=
a


m=1



b
m1



=

Km (x, u)ai (u)du .
a

m=1

G(x, ; t) õ t ữủ t t ữ s
n

[ai (x) + Ai (x; )]bi (t).

G (x, t; ) =



i=1

ữ ữỡ tr tr t ữỡ tr t r ợ
t õ t ữỡ tr ữỡ ữủ ổ t
ữỡ õ sỹ t õ ỏ ử tở t số

. s qt ữỡ tr ữ s
ừ G(x, t; ) tr ữỡ tr t ữủ
n

(x) = f (x; ) +

ci ()[ai (x) + Ai (x; )],



i=1

tr õ t t
b

(t)bi (t)dt,

ci () =

i = 1, ..., n.

a

sỷ r ộ ừ ữỡ tr õ t t
số ci(). r ữỡ tr t t t x t ờ số ừ tờ tứ i t
j, s õ ừ ữỡ tr ợ bi (t) rỗ t tứ a tợ b t t ữủ
b

(t)bi (t)dt =
a


n

b

cj ()

f (x; )bi (t)dt +
a

b

[aj (t) + Aj (t; )]bi (tdt.
a

j=1

t
b

fi () =

f (x; )bi (t)dt,
a
b

aij () =

[aj (t) + Aj (t; )]bi (tdt.
a


õ t ữủ ữỡ tr
n

ci () = fi () +

cj ()aij ().
j=1






❍❛② t❛ ❝ô♥❣ ❝â t❤➸ ✈✐➳t ❧↕✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ tr➯♥ ❞÷î✐ ❞↕♥❣ ♠❛ tr➟♥ ♥❤÷ s❛✉
(■ − λ❆(λ))❝(λ) = ❢(λ).
✭✷✳✶✻✮
✣➦t
Dρ (λ) = ❞❡t(■ − λ❆(λ)),

✈➔ ❣å✐ Dρ(λ) ❧➔ ✤à♥❤ t❤ù❝ ❋r❡❞❤♦❧♠ ✳ ❚❛ t❤➜② Dρ(λ) ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ ❝õ❛ λ tr➯♥ ✤➽❛ ✤â♥❣
ρ . ❑❤✐ ✤â sè ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✻✮ ♣❤ö t❤✉ë❝ ✈➔♦ ❣✐→ trà ❝õ❛ Dρ (λ). ❚❛
s➩ ①❡♠ ①➨t ❤❛✐ tr÷í♥❣ ❤ñ♣ s❛✉✿
❚r÷í♥❣ ❤ñ♣ ✶✿ Dρ(λ) = 0. ❑❤✐ ✤â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ tr➯♥ ❝â ♥❣❤✐➺♠ ❞✉②
♥❤➜t

❝(λ)

= (■ − λ❆(λ))−1 ❢(λ)
1

=
❛❞❥(■ − λ❆(λ))❢(λ)
Dρ (λ)

tr♦♥❣ ✤â ❛❞❥(■ − λ❆(λ)) = (Dji(λ)) ❧➔ ♠❛ tr➟♥ ♣❤ö ❤ñ♣ ❝õ❛ ■ − λ❆(λ). ▼é✐ ❤➺ sè ci(λ)
❝â t❤➸ ✤÷ñ❝ ❜✐➸✉ ❞✐➵♥ ❞÷î✐ ❞↕♥❣
1
ci (λ) =
Dρ (λ)

n

Dji (λ)fj (λ).
j=1

✭✐✮ ◆➳✉ ❢(λ) = 0 t❤➻ ❝(λ) = 0✱ ♥❣❤➽❛ ❧➔ ci(λ) = 0 ✈î✐ ♠å✐ i = 1, ..., n. ❉♦ ✤â tø
♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✹✮ s✉② r❛ ϕ(x) = fε(x; λ).
✭✐✐✮ ◆➳✉ ❢(λ) = 0 t❤➻ ❝(λ) = 0, ♥❣❤➽❛ ❧➔ tç♥ t↕✐ ➼t ♥❤➜t ♠ët ❝❤➾ sè ❞÷î✐ i s❛♦ ❝❤♦
ci (λ) = 0. ❚❤❛② ❝→❝ ❣✐→ trà ci (λ) ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✹✮ t❛ t❤✉ ✤÷ñ❝
n

ϕ(x) = fε (x; λ) + λ
i=1
b

= fε (x; λ) + λ

1
Dρ (λ)


n

Dji (λ)fj (λ) [ai (x) + λAεi (x; λ)]
j=1

Sε (x, t; λ)fε (t; λ)dt,
a

tr♦♥❣ ✤â t❛ ✤➦t
n

n

Dji (λ)[ai (x) + λAεi (x; λ)]bj (t)
Sε (x, t; λ) =

i=1 j=1

Dρ (λ)

.

✭✷✳✶✼✮

◆❤➙♥ Sε(x, t; λ) ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ ❝õ❛ λ tr➯♥ ✤➽❛ ✤â♥❣ ρ. ❍ì♥ ♥ú❛ ♥â ❝á♥ ❧➔ ❤➔♠ t→❝❤
❜✐➳♥ ✤è✐ ✈î✐ ❤❛✐ ❤➔♠ x ✈➔ t. ❚❤❛② fε(x; λ) ✈➔♦ tr♦♥❣ ❜✐➸✉ ❞✐➵♥ tr➯♥ ❝õ❛ ϕ(x) t❛ t❤✉ ✤÷ñ❝
✤↕♥❣ t❤✉ ❣å♥ ❝õ❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✵✮ ✈➔ ❝ô♥❣ ❝❤➼♥❤ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✻✮
♥❤÷ s❛✉
b


ϕ(x) = f (x) + λ

tr♦♥❣ ✤â

Uε (x, t; λ)f (t)dt
a

b

Uε (x, t; λ) = Rε (x, t; λ) + Sε (x, t; λ) + λ

Sε (x, t; λ)Rε (s, t; λ)ds.
a

✷✶

✭✷✳✶✽✮


ứ t t R(x, t; ) tữỡ ố ọ õ tứ t t S(x, t; ) t
ừ U(x, t; ). S(x, t; ) U(x, t; ) ụ
tr . ỡ ỳ U(x, t; ) ỏ tr t t ự
sỷ ởt số tũ ỵ s 0 < < . õ tữỡ tỹ t ụ
ỹ ữủ t tỷ U(x, t; ) tr . õ ừ ữỡ
tr õ
b

(x) = f (x) +

U (x, t; )f (t)dt.

a

ữ t U(x, t; ) = U(x, t; ) tr . ữ U(x, t; )
t tr tứ U(x, t; ). õ t ợ tũ ỵ s r U(x, t; )
õ t ữủ t tr t ởt tr t t ự
sỹ tr t R(x, t; ) ồ t tỷ .
ứ ự tr t t ữủ t q s

ỵ ỵ r tự t t ữỡ tr t r



b

(x) = f (x) +

K(x, t)(t)dt,
a

tr õ ởt t số ự f (x) C[a, b], K(x, t) C(Q[a, b]). ởt
tr q ừ K(x, t) t ữỡ tr tr õ t ữủ
ổ tự
b

(x) = f (x) +

R(x, t; )f (t)dt,
a

t tỷ t R(x, t; ) ữủ ữ tr


ỵ ỵ r tự tữ t ữỡ tr t t t
b

K(x, t)(t)dt,

(x) =
a

tr õ K(x, t) C(Q[a, b]). K t tr r ừ K(x, t).
õ K ỳ ữủ ổ õ ợ ỳ

tự D() ởt t ừ tr õ . s ự
K ỳ ợ ồ > 0. sỷ ự r K ổ õ
D() õ ổ ổ tr . D() t t D() 0
tr . ổ D(0) = 1. õ tỗ t ỳ tr tr s
D() = 0.
rữớ ủ D() = t( ()) = 0. r trữớ ủ õ ỳ
tr tở s D() = 0. t s
f () = 0 t t t
ự

( ())()

õ p() tỡ ở t t c(j)(), j = 1, 2, ..., p()
õ ữủ t ữợ


c(j) () =


(1)

cj ()



(n)
cj ()







,

j = 1, ..., p().


❚❤❛② ❝→❝ ❣✐→ trà ❝õ❛ cji (λ) ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✹✮ t❛ t❤✉ ✤÷ñ❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
✭✷✳✻✮✳ ◆➳✉ f (x) ≡ 0 tr➯♥ [a, b] t❤➻ fε(x; λ) ≡ 0. ❑❤✐ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✻✮ ✤÷ñ❝ t❤✉ ❣å♥ t❤➔♥❤
n
(e)
ϕj (x; λ)

(j)

=


ci (λ) [ai (x) + λAεi (x; λ)]
i=1
n
(j)
ci (λ)

=

b

ai (x) + λ

Rε (x, t; λ)ai (u)du ,

j = 1, ..., p(λ).

a

i=1

❈❤➾ sè tr➯♥ (e) ✤➸ ❦➼ ❤✐➺✉ r➡♥❣ ϕj(e)(x; λ) ❧➔ ♠ët ❤➔♠ r✐➯♥❣ ❝õ❛ ♥❤➙♥ K(x, t). ◆➳✉ t❛ ❦➼
❤✐➺✉ ϕh(x; λ) ❧➔ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t❤✉➛♥ ♥❤➜t K(x, t) t❤➻ ♥â
❝â ❞↕♥❣
p
(e)

h

(λ)αj ϕj (x; λ),


ϕ (x; λ) =
j=1

tr♦♥❣ ✤â αj ❧➔ ❤➡♥❣ sè tò② þ✱ ❝❤➾ sè tr➯♥ (h) ✤➸ ❦➼ ❤✐➺✉ r➡♥❣ ϕh(x; λ) ❧➔ ♥❣❤✐➺♠ tê♥❣
q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ ✈î✐ ❣✐→ trà r✐➯♥❣ λ. ❚÷ì♥❣ tü ♥❤÷ ð ❈❤÷ì♥❣
✶✱ t❛ s➩ ✤✐ ①❡♠ ①➨t ♠è✐ ❧✐➯♥ ❤➺ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✻✮ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❧✐➯♥
❦➳t tr♦♥❣ tr÷í♥❣ ❤ñ♣ λ ❧➔ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♥❤➙♥ tê♥❣ q✉→t✳
❳➨t ♣❤÷ì♥❣ tr➻♥❤
b
K(t, x)ψ(t)dt
✭✷✳✶✾✮
ψ=λ
a

tr♦♥❣ ✤â K(t, x) t❤✉ë❝ C(Q[a, b]). ❚ø ✭✷✳✼✮ t❛ s✉② r❛
K(t, x) = Ksep (t, x) + Kε (t, x).

❚❤❛② K(t, x) ✈➔♦ ✭✷✳✶✾✮ t❛ ✤÷ñ❝
b

ω(x) = ψ(x) − λ

b

Kε (t, x)ψ(t)dt = λ
a

Ksep (t, x)ψ(t)dt
a


✭✷✳✷✵✮

❚❛ t❤➜② ♣❤÷ì♥❣ tr➻♥❤
b

ψ(x) = ω(x) + λ

Kε (t, x)ψ(t)dt
a

❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ✈î✐ ♥❤➙♥ ❝â ❝❤✉➞♥ ✤õ ♥❤ä✱ ❞♦ ✈➟② →♣ ❞ö♥❣ ✤à♥❤ ❧þ ①➜♣
①➾ ❧✐➯♥ t✐➳♣ t❛ t❤✉ ✤÷ñ❝ ♥❣❤✐➺♠ ψ(x) ❝â ❞↕♥❣
b

ψ(x) = ω(x) + λ

Rε (t, x; λ)ω(t)dt,
a

✭✷✳✷✶✮

tr♦♥❣ ✤â Rε(t, x; λ) ❧➔ ♥❤➙♥ ❣✐↔✐ ❧✐➯♥ ❤ñ♣ ✈î✐ ❤↕t ♥❤➙♥ Rε(t, x; λ) ✤÷ñ❝ ①➙② ❞ü♥❣ tø ❝→❝
❤↕t ♥❤➙♥ ❧➦♣ ❝õ❛ ♥❤➙♥ Kε(x, t). ❚❤❛② ω(x) = λ ab Ksep(t, x)ψ(t)dt t❛ ✤÷ñ❝
b

ψ(x) = λ

Gε (t, x; λ)ψ(t)dt,
a


✷✸

✭✷✳✷✷✮


tr♦♥❣ ✤â Gε(t, x; λ) ❧➔ ❧✐➯♥ ❤ñ♣ ♣❤ù❝ ❝õ❛ ♥❤➙♥ Gε(t, x; λ) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✵✮✳ ❙✉② r❛
n

Gε (t, x; λ) =

[ai (t) + λAεi (t; λ)]bi (x).

✭✷✳✷✸✮

i=1

❚❛ t❤➜② Gε(t, x; λ) ❝â ❞↕♥❣ t→❝❤ ❜✐➳♥ ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✷✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥
t❤✉➛♥ ♥❤➜t ✈î✐ ♥❤➙♥ t→❝❤ ❜✐➳♥✳ ❱➻ ✈➟②✱ ♥❣❤✐➺♠ ❝õ❛ ♥â ❝â ❞↕♥❣
n
(e)
ψj (x, λ)

(j)

di (λ)bi (x),

=λ

j = 1, .., q(λ).


i=1

tr♦♥❣ ✤â q(λ) ❧➔ sè ❤➔♠ r✐➯♥❣ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ t÷ì♥❣ ù♥❣ ✈î✐ λ ✈➔
(j)
di (λ)

b

ψ(t)[ai (t) + λAεi (t; λ)]bi (x).

=
a

❉♦ ✤â λ ❧➔ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♥❤➙♥ K(t, x) t÷ì♥❣ ù♥❣ ✈î✐ q(λ) ❤➔♠ r✐➯♥❣ ✤➣ ❝❤♦✳ ▲↕✐ ❝â
♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✷✮ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ϕ(x) = ab Gε(x, t; λ)ϕ(t)dt ❝â ❤↕t ♥❤➙♥ ❧✐➯♥ ❦➳t
♥➯♥ ❝❤ó♥❣ ♣❤↔✐ ❝â ❝ò♥❣ sè ❤➔♠ r✐➯♥❣ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✱ ♥❣❤➽❛ ❧➔ p(λ) = q(λ). ❚ø ✤â✱
t❛ ❝â ✤à♥❤ ❧þ s❛✉✿

✣à♥❤ ❧þ ✷✳✸✳✸✳ ✭✣à♥❤ ❧þ ❋r❡❞❤♦❧♠ t❤ù ❤❛✐✮✳ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t❤✉➛♥ ♥❤➜t
b

ϕ(x) = λ

K(x, t)ϕ(t)dt.
a

tr♦♥❣ ✤â λ ❧➔ ♠ët t❤❛♠ sè ♣❤ù❝✱ f (x) ∈ C[a, b] ✈➔ K(x, t) ∈ C(Q[a, b]). ◆➳✉ λ ❧➔ ♠ët
❣✐→ trà r✐➯♥❣ ❝õ❛ ♥❤➙♥ K(x) t❤➻ λ ❧➔ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♥❤➙♥ ❧✐➯♥ ❤ñ♣ K(x, t) ✈➔ sè ❝→❝
❤➔♠ r✐➯♥❣ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❜➡♥❣ ✈î✐ sè ❤➔♠ r✐➯♥❣ ❝õ❛

♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t ❧✐➯♥ ❦➳t
b

ϕ(x) = λ

K(t, x)ψ(t)dt
a

✭✐✐✮ ❢(λ) = 0. ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤
b

ϕ(x) = fε (x; λ) + λ

Gε (x, t; λ)ϕ(t)dtdx
a

✭✷✳✷✹✮

❝â Gε(x, t; λ) ❧➔ ❤↕t ♥❤➙♥ t→❝❤ ❜✐➳♥✳ ❱➻ ✈➟②✱ t❤❡♦ ✤à♥❤ ❧þ ❋r❡❞❤♦❧♠ t❤ù ❜❛✱ ♣❤÷ì♥❣ tr➻♥❤
✭✷✳✷✹✮ ❝â ♥❣❤✐➺♠ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ fε(x; λ) trü❝ ❣✐❛♦ ✈î✐ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
t❤✉➛♥ ♥❤➜t ❧✐➯♥ ❦➳t
b

ψ(x) = λ

Gε (x, t; λ)ψ(t)dt.
a

◆➳✉ ω(x) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t tr➯♥ t❤➻ tø ❜✐➸✉ ❞✐➵♥ ✭✷✳✶✶✮ ✈➔


✷✹


✭✷✳✷✶✮ t❛ ❝â
b

b

fε (t; λ)ω(t)dt =

b

Rε (t, s; λ)f (s)ds ω(t)dt

f (t) + λ
a

a

a
b

=

b

b

f (t)ω(t)dt + λ
a

b

=

Rε (t, s; λ)f (s)ω(t)dsdt
a
b

f (t) ω(t) + λ
a

a

Rε (s, t; λ)ω(t)ds dt
a

b

=

b

f (t) ω(t) + λ
a

Rε (s, t; λ)ω(s)ds dt
a

b


=

f (t)ψ(t)dt.
a

❚ø ✤â t❛ t❤➜② r➡♥❣ fε(x; λ) trü❝ ❣✐❛♦ ✈î✐ ω(x) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f (x) trü❝ ❣✐❛♦ ✈î✐ ♥❣❤✐➺♠
ψ(x) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❧✐➯♥ ❦➳t
b

ψ(x) = λ

K(t, x)ψ(t)dt.
a

❚❛ ❝â ✤à♥❤ ❧þ s❛✉✿

✣à♥❤ ❧þ ✷✳✸✳✹✳ ✭✣à♥❤ ❧þ ❋r❡❞❤♦❧♠ t❤ù ❜❛✮✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐
b

ϕ(x) = f (x) + λ

K(x, t)ϕ(t)dt,
a

tr♦♥❣ ✤â λ ❧➔ ♠ët t❤❛♠ sè ♣❤ù❝✱ f (x) ∈ C[a, b] ✈➔ K(x, t) ∈ C(Q[a, b]). ◆➳✉ λ ❧➔ ♠ët
❣✐→ trà r✐➯♥❣ ❝õ❛ ♥❤➙♥ K(x, t) t❤➻ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ tr➯♥ ❝â ♥❣❤✐➺♠ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
f (x) trü❝ ❣✐❛♦ ✈î✐ t➜t ❝↔ ❝→❝ ❤➔♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❧✐➯♥ ❦➳t
b

ψ(x) = λ


K(x, t)ψ(t)dt.
a

✷✳✹ ❈➜✉ tró❝ ❝õ❛ ♥❤➙♥ ❣✐↔✐
Ð ♣❤➛♥ ♥➔②✱ t❛ s➩ ✤✐ ①➙② ❞ü♥❣ ❝➜✉ tró❝ ❝õ❛ ♥❤➙♥ ❣✐↔✐ ❦❤✐ λ ❧➔ ❣✐→ trà ❝❤➼♥❤ q✉②✳ ❚❛
t❤✉ ✤÷ñ❝ ❝→❝ ❦➳t q✉↔ ❞÷î✐ ✤➙②✿

✣à♥❤ ❧þ ✷✳✹✳✶✳ ◆❤➙♥ ❣✐↔✐ R(x, t; λ) ❝â t❤➸ ✤÷ñ❝ ❜✐➸✉ ❞✐➵♥ ♥❤÷ ❧➔ t❤÷ì♥❣ ❝õ❛ ❤➔♠ ♥❣✉②➯♥

D(x, t; λ) ✈➔ D(λ) ✤➣ ❝❤♦ ❜ð✐ ❝→❝ ❝❤✉é✐ ❧ô② t❤ø❛
∞

D(x, t; λ) =
i=1

(−1)n
Bn (x, t)λn
n!

∞

✈➔ D(λ) =
i=1

(−1)n
cn λn ,
n!

tr♦♥❣ ✤â B0 (x, t) = K(x, t) ✈➔ c0 = 1. ❱î✐ n ≥ 1 t❤➻

b

b

Bn−1 (t, t)dt ✈➔ Bn (x, t) = cn K(x, t) − n

cn =
a

K(x, s)Bn−1 (s, t)dt.
a

✷✺