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à §àë¢ë ⮫쪮 ¯¥à¢®£® த  ¨

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(0.6)

a

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(

x;u

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(

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(

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g

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2

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(

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u

, ¤«ï ª®â®àëå

g

(

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> g

(

u

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2

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u

(

x

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g

;(

x;u

(

x

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+(

x;u

(

x

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x

2, £¤¥



| ä®à¬ «ì­® á ¬®á®¯à殮­­ë©, à ¢­®¬¥à­® í««¨¯â¨ç¥áª¨©, «¨­¥©-­ë© ¤¨ää¥à¥­æ¨ «ì«¨­¥©-­ë© ®¯¥à â®à ¯®à浪  2

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(

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R

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dx

Z u (x) 0

g

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u

(

x

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g

(

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;g

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u

(

x

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u

(

x

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g

(

x;u

(

x

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x

2 . ‘ãâì ¥£® ®£à ­¨ç¥­¨© ¢ ⮬, çâ® ¢á¥ à §àë¢ë

g

(

x;u

) ¯®

u

«¥¦ â ­  ­¥ ¡®«¥¥ 祬 áç¥â­®¬ ¬­®¦¥á⢥ ¤®áâ â®ç­® £« ¤ª¨å ¯®¢¥àå­®á⥩ ¨, ¥á«¨

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(

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(

x

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=

[

g

;(

x;'

(

x

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;g

+(

x;'

(

x

))]. ‚ ¤ ­­®© à ¡®â¥ ¤«ï  ¡áâà ªâ­ëå ãà ¢­¥­¨© á à §à뢭묨 ­¥ ª®íàæ¨â¨¢­ë¬¨ ®¯¥à â®à ¬¨ ¯®«ãç¥­ë ­®¢ë¥ ¢ à¨ æ¨®­­ë¥ ¯à¨­æ¨¯ë áãé¥á⢮¢ ­¨ï à¥è¥­¨©, ª®â®àë¥ ï¢«ïîâáï â®çª ¬¨ ­¥¯à¥à뢭®á⨠®¯¥à â®à  ãà ¢­¥­¨ï. Ž¡é¨¥ १ã«ìâ âë ¯à¨¬¥­ïîâáï § â¥¬ ª ¨§ã祭¨î § ¤ -ç¨ (0.1){(0.2) ¢ १®­ ­á­®¬ á«ãç ¥. „®ª §ë¢ îâáï ¯à¥¤«®¦¥­¨ï ⨯  ‹ ­¤¥­á¬ ­ -‹ §¥à  ® áãé¥á⢮¢ ­¨¨ ᨫì­ëå ¨ ¯®«ã¯à ¢¨«ì­ëå à¥è¥­¨© (ᨫ쭮¥ à¥è¥­¨¥ § ¤ ç¨ (0.1){(0.2) ­ §ë-¢ ¥âáï ¯®«ã¯à ¢¨«ì­ë¬, ¥á«¨ ¤«ï ¯®ç⨠¢á¥å

x

2 §­ ç¥­¨¥

u

(

x

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(

x;

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u

­¥«¨­¥©­®áâìî ¡ë«¨ ¢¢¥¤¥­ë ¢ [11]. ‚®¯à®á ® áãé¥á⢮¢ ­¨¨ ¯®«ã¯à ¢¨«ì­ëå à¥-襭¨© ãà ¢­¥­¨ï (0.1), 㤮¢«¥â¢®àïîé¨å ®¤­®à®¤­®¬ã £à ­¨ç­®¬ã ãá«®¢¨î „¨à¨å«¥, ¨§ãç «áï ¢ [12], £¤¥ ¯à¥¤¯®« £ «®áì, çâ® ­¥«¨­¥©­®áâì

g

(

x;u

) ®£à ­¨ç¥­  ­  

R

¨ ¯®

u

㤮¢«¥â¢®àï¥â

®¤­®áâ®à®­­¥¬ã ãá«®¢¨î ‹¨¯è¨æ , ª®â®à®¥ ¢«¥ç¥â ­¥à ¢¥­á⢮

g

(

x;u

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(

x;u

+) ¤«ï «î¡®-£®

u

2

R

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x

2 . Žâ¬¥â¨¬, çâ® ¢ [12]  ¢â®àë ­ §ë¢ îâ ¯®«ã¯à ¢¨«ì­ë¥ à¥è¥­¨ï ¯à ¢¨«ì­ë¬¨. ® áà ¢­¥­¨î á à ¡®â ¬¨ ¤àã£¨å  ¢â®à®¢ ¯® ¯à®¡«¥¬¥ áãé¥á⢮¢ ­¨ï ᨫì­ëå à¥è¥­¨© § ¤ ç¨ (0.1){(0.2) ¢ १®­ ­á­®¬ á«ãç ¥ ¢ ¤ ­­®© áâ âì¥ ®á« ¡«¥­ë ®£à ­¨ç¥­¨ï ­  ¬­®¦¥-á⢮ â®ç¥ª à §à뢠 ­¥«¨­¥©­®áâ¨

g

(

x;u

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u

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g

(

x;u

) ¯®

u

, ¤«ï ª®â®àëå

g

(

x;u

;)

> g

(

x;u

+) (\¯ ¤ î騥 à §àë¢ë"),   ¢ ®â«¨ç¨¥ ®â [12] ¤®¯ã᪠îâáï à §àë¢ë, \¯àë-£ î騥 ¢¢¥àå" (

g

(

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< g

(

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+)). ‘âpãªâãà  áâ âì¨ á«¥¤ãîé ï. ‚ ¯¥à¢®¬ ¯ã­ªâ¥ ¯à¨¢®¤ïâáï ä®à¬ã«¨à®¢ª¨ ®á­®¢­ëå १ã«ì-â â®¢ à ¡®âë. ‚® ¢â®à®¬ ¤®ª §ë¢ îâáï ®¡é¨¥ ¢ à¨ æ¨®­­ë¥ ¯à¨­æ¨¯ë. ‚ âà¥â쥬 à áᬠâà¨-¢ îâáï ¯à¨«®¦¥­¨ï ®¡é¨å ⥮६ ª ªà ¥¢ë¬ § ¤ ç ¬ ¤«ï ãà ¢­¥­¨© í««¨¯â¨ç¥áª®£® ⨯  á à §à뢭묨 ­¥«¨­¥©­®áâﬨ (ãáâ ­ ¢«¨¢ îâáï ¯à¥¤«®¦¥­¨ï ⨯  ‹ ­¤¥á¬ ­ -‹ §¥à  [3]). 1. ”®à¬ã«¨à®¢ª  ®á­®¢­ëå १ã«ìâ â®¢ ãáâì

X

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X

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y

2

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2

Y

| h

y;x

i.  áᬠâਢ îâáï ãà ¢­¥­¨ï ¢¨¤ 

Qx



Ax

+

P



TPx

;

p

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;

(1.1) £¤¥

A

| «¨­¥©­ë© ®£à ­¨ç¥­­ë© á ¬®á®¯à殮­­ë© ®¯¥à â®à ¢

X

á ­¥­ã«¥¢ë¬ ï¤à®¬

N

(

A

), ®â®¡à ¦¥­¨¥

T

:

Y

!

Y

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Y

(¢®§¬®¦­®, à §à뢭®¥),

p

2

X

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T

:

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!

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f

:

Y

!

R

, ¤«ï ª®â®à®£®

f

(

x

+

h

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f

(

x

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T

(

x

+

th

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dt

8

x;h;

2

Y:

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f

­ §ë¢ îâ ª¢ §¨¯®â¥­æ¨ «®¬ ®¯¥à â®à 

T

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x

2

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­ §ë¢ ¥âáï ॣã«ïà­ë¬ [14] (ᨫ쭮 ॣã«ïà­ë¬) ¤«ï ®¯¥à â®à 

Q

:

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h

2

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â ª®©, çâ® limsup_t !+0 h

Q

(

x

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2

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:

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x

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@'

(

x

0) { ®¡®¡é¥­­ë© £à ¤¨¥­â Š« àª  ä㭪樨

'

¢ â®çª¥

x

0 ([15], c.34),   á ¬ã

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0 | ªà¨â¨ç¥áª®© â®çª®© ä㭪樮­ « 

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­ §ë¢ ¥âáï â®çª®© à §à뢠 ®¯¥à â®à 

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:

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0 ­ àã襭® ãá«®¢¨¥ à ¤¨ «ì­®© ­¥¯à¥à뢭®á⨠¤«ï ®¯¥à â®à 

Q

([16], c.79): lim t!0 h

Q

(

x

0+

th

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;h

i =h

Qx

0

;h

i 8

h

2

X:

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¢

Y

; 2) ®¯¥à â®à

A

:

X

!

X

«¨­¥©­ë©, ®£à ­¨ç¥­­ë© ¨ á ¬®á®¯à殮­­ë©, ­ã«ì ï¥âáï ¨§®-«¨à®¢ ­­®© â®çª®© ¥£® ᯥªâà , ¯à¨ç¥¬ (

Ax;x

)0 8

x

2

X

;

3) ®â®¡à ¦¥­¨¥

T

:

Y

!

Y

 ª¢ §¨¯®â¥­æ¨ «ì­®¥ ¨ ®£à ­¨ç¥­­®¥ ­ 

Y

, â.¥. áãé¥áâ¢ã¥â ª®­áâ ­â 

M >

0, ¤«ï ª®â®à®© k

Tx

k

M

8

x

2

Y

; 4) í«¥¬¥­â

p

2

X

㤮¢«¥â¢®àï¥â ãá«®¢¨î lim x2N(A);kxk!+1 (

f

(

x

);(

p;x

)) = +1

;

(1.3) £¤¥

f

| ª¢ §¨¯®â¥­æ¨ « ®¯¥à â®à 

T

. ’®£¤  áãé¥áâ¢ã¥â

x

0 2

X

, ¤«ï ª®â®à®£®

'

(

x

0) = inf X

'

(

x

),

'

(

x

) = (

Ax;x

)

=

2+

f

(

x

);(

p;x

), ¯à¨ç¥¬ «î¡®¥ â ª®¥

x

0 㤮¢«¥â¢®àï¥â ¢ª«î祭¨î ;

Ax

0+ 

p

2

P

 (

ST

)(

Px

0)

;

(1.4) £¤¥

ST

| ᥪ¢¥­æ¨ «ì­®¥ § ¬ëª ­¨¥ ®¯¥à â®à 

T

[17]. …᫨ ª ⮬㠦¥ ¢á¥ â®çª¨ à §à뢠 ®¯¥-à â®à (1

:

1) ॣã«ïà­ë¥ ¤«ï

Q

, â® «î¡®¥ â ª®¥

x

0 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î (1

:

1) ¨ ï¥âáï â®çª®© à ¤¨ «ì­®© ­¥¯à¥à뢭®á⨠®¯¥à â®à 

P



TP

. ‡ ¬¥ç ­¨¥ 1.1. “á«®¢¨¥ 3) ⥮६ë 1.1 ¢«¥ç¥â «¨¯è¨æ¥¢®áâì ­ 

Y

ª¢ §¨¯®â¥­æ¨ « 

f

®¯¥-à â®à 

T

. „¥©á⢨⥫쭮, ¤«ï ¯à®¨§¢®«ì­ëå

u;v

2

Y

¨¬¥¥¬ j

f

(

u

);

f

(

v

)j=     Z 1 0 h

T

(

v

+

t

(

u

;

v

))

;u

;

v

i

dt

     Z 1 0 jh

T

(

v

+

t

(

u

;

v

))

;u

;

v

ij

dt



M

k

u

;

v

kY

;

£¤¥

M

| ª®­áâ ­â  ¨§ ãá«®¢¨ï 3) ⥮६ë 1.1. “ª ¦¥¬ ­  ¯à®á⮥ ¤®áâ â®ç­®¥ ãá«®¢¨¥ ॣã«ïà­®á⨠â®ç¥ª à §à뢠 ®¯¥à â®à  (1.1). à¥¤«®¦¥­¨¥ 1.1. …᫨

x

| â®çª  à §à뢠 ®¯¥à â®à 

P



TP

¨ áãé¥áâ¢ã¥â lim t!+0 h

T

(

x

+

th

)

;h

ih

Tx;h

i 8

h

2

X

, â®

x

| ॣã«ïà­ ï â®çª  ¤«ï ®¯¥à â®à  (1

:

1). „¥©á⢨⥫쭮, ¢ í⮬ á«ãç ¥ áãé¥áâ¢ã¥â lim_t !+0 h

Q

(

x

+

th

)

;h

ih

Qx;h

i8

h

2

X

¨, ¥á«¨ ¯à¥¤¯®-«®¦¨âì, çâ® ¢ â®çª¥

x

­ àã襭® ãá«®¢¨¥ (1.2) à ¤¨ «ì­®© ­¥¯à¥à뢭®áâ¨, â® ­ ©¤¥âáï

h

2

X

, ¤«ï ª®â®à®£® lim_t !+0 h

Q

(

x

+

th

)

;h

i

<

h

Qx;h

i. ®í⮬㠯à¨h

Qx;h

i 0 ॣã«ïà­®áâì

x

¤«ï

Q

¤®ª -§ ­ . ‚ ¯à®â¨¢­®¬ á«ãç ¥h

Qx;h

i

>

0. ’®£¤  lim t!+0 h

Q

(

x

+

t

(;

h

))

;

(;

h

)i h

Qx;

;

h

i=;h

Qx;h

i

<

0, ¨ §­ ç¨â,

x

| ॣã«ïà­ ï ¤«ï

Q

â®çª 

:

 ‚® ¢â®à®¬ ¢ à¨ æ¨®­­®¬ ¯à¨­æ¨¯¥ (⥮६  1.2) â®çª ¬¨ à §à뢠 ®¯¥à â®à 

Q

:

X

!

X

 ¡ã¤¥¬ ­ §ë¢ âì â¥

x

2

X

, ¢ ª®â®àëå ­ àã襭® ãá«®¢¨¥ ¤¥¬¨­¥¯à¥à뢭®á⨠¤«ï ®¯¥à â®à 

Q

([13], c.23).  ¯®¬­¨¬, çâ® ¥á«¨ ­ã«ì ï¥âáï ¨§®«¨à®¢ ­­®© â®çª®© ᯥªâà  «¨­¥©­®£® ®£à -­¨ç¥­­®£® ¨ á ¬®á®¯à殮­­®£® ®¯¥à â®à 

A

¢ £¨«ì¡¥à⮢®¬ ¯à®áâà ­á⢥

X

, â®

X

à á¯ ¤ ¥âáï ­  á㬬㠮à⮣®­ «ì­ëå ¨ ¨­¢ à¨ ­â­ëå ¯® ®â­®è¥­¨î ª

A

¯®¤¯à®áâà ­áâ¢:

N

(

A

)

; X

+= f

x

2

X

j(

Ax;x

)

>

0g[f0g

; X

; = f

x

2

X

j(

Ax;x

)

<

0g[f0g

:

®¤¯à®áâà ­á⢠

X

;,

X

+ ­ §ë¢ îâáï ᮮ⢥âá⢥­­® ®âà¨æ â¥«ì­ë¬ ¨ ¯®«®¦¨â¥«ì­ë¬ ¯®¤-¯à®áâà ­á⢠¬¨ ®¯¥à â®à 

A

. ’¥®à¥¬  1.2. à¥¤¯®«®¦¨¬, çâ® 1) ¢ë¯®«­¥­ë ãá«®¢¨ï 1), 3) ⥮६ë 1

:

1 ¨

X

¯«®â­® ¢

Y

; 2) ®¯¥à â®à

A

:

X

!

X

«¨­¥©­ë©, ®£à ­¨ç¥­­ë© ¨ á ¬®á®¯à殮­­ë©, ­ã«ì ï¥âáï ¨§®-«¨à®¢ ­­®© â®çª®© ¥£® ᯥªâà , ¯à¨ç¥¬ ï¤à®

N

(

A

) ¨ ®âà¨æ â¥«ì­®¥ ¯®¤¯à®áâà ­á⢮

X

; ®¯¥à â®à 

A

ª®­¥ç­®¬¥à­ë; 3) í«¥¬¥­â

p

2

X

㤮¢«¥â¢®àï¥â ãá«®¢¨î lim x2N(A); kxk!+1 (

f

(

x

);(

p;x

)) = +1 ¨«¨ ;1

;

(1.5) £¤¥

f

| ª¢ §¨¯®â¥­æ¨ « ®¯¥à â®à 

T

.

’®£¤  ¬­®¦¥á⢮ ªà¨â¨ç¥áª¨å â®ç¥ª ä㭪樮­ « 

'

(

x

) = (

Ax;x

)

=

2 +

f

(

x

);(

p;x

) ­¥ ¯ãáâ® ¨ ª ¦¤ ï â®çª 

x

¨§ í⮣® ¬­®¦¥á⢠ 㤮¢«¥â¢®àï¥â ¢ª«î祭¨î (1

:

4). …᫨ ª ⮬㠦¥ ¢á¥ â®çª¨ à §à뢠 ®¯¥à â®à  (1

:

1) ᨫ쭮 ॣã«ïà­ë¥, â® «î¡ ï ªà¨â¨ç¥-᪠ï â®çª  ä㭪樮­ « 

'

㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î (1

:

1) ¨ ï¥âáï â®çª®© ¤¥¬¨­¥¯à¥àë¢-­®á⨠®¯¥à â®à 

Q

. à¨¬¥­¥­¨¥ áä®à¬ã«¨à®¢ ­­ëå ¢ëè¥ ¢ à¨ æ¨®­­ëå ¯à¨­æ¨¯®¢ ª § ¤ ç¥ (0.1){(0.2) ¤ ¥â á«¥¤ãî騥 १ã«ìâ âë. Ž¯à¥¤¥«¥­¨¥ 1.3. ƒ®¢®àïâ, çâ® ¤«ï ãà ¢­¥­¨ï (0.1) ¢ë¯®«­¥­®

A

-ãá«®¢¨¥ |

Ay

[18] (ᨫì-­®¥

Ay

), ¥á«¨ ­ ©¤¥âáï ­¥ ¡®«¥¥ 祬 áç¥â­®¥ ᥬ¥©á⢮ ¯®¢¥àå­®á⥩ f

S

i

; i

2

I

g,

S

i =f(

x;u

) 2

R

n+1 j

u

=

'

i(

x

)

; x

2 g,

'

i 2

W

2 loc;1() â ª¨å, çâ® ¤«ï ¯®ç⨠¢á¥å

x

2 ­¥à ¢¥­á⢮

g

(

x;u

;)

< g

(

x;u

+) (

g

(

x;u

;)=6

g

(

x;u

+)) ¢«¥ç¥â áãé¥á⢮¢ ­¨¥

i

2

I

, ¤«ï ª®â®à®£®

u

=

'

i(

x

) ¨ (

L'

i(

x

) +

g

(

x;'

i(

x

)+);

p

(

x

))(

L'

i(

x

) +

g

(

x;'

i(

x

););

p

(

x

))

>

0

:

(1.6) Ž¯à¥¤¥«¥­¨¥ 1.4. ƒ®¢®àïâ,çâ® ¤«ï ãà ¢­¥­¨ï (0.1) ¢ë¯®«­¥­®

A

1-ãá«®¢¨¥ |

A

1

y

(ᨫì-­®¥

A

1

y

), ¥á«¨ 㤮¢«¥â¢®àï¥âáï ®¯à¥¤¥«¥­¨¥ 1.3, ¢ ª®â®à®¬ ¢¥à­® «¨¡® (1.6), «¨¡®

L'

i(

x

) +

g

(

x;'

i(

x

)) =

p

(

x

). ‘®¯®áâ ¢¨¬ ªà ¥¢®© § ¤ ç¥ (0.1){(0.2) ¯à¨ 䨪á¨à®¢ ­­®¬

p

2

L

q(),

q >

2

n=

(

n

+ 2), äã­ª-樮­ «

J

p :

X

!

R

, £¤¥

X

= 

W

1 2() ¢ á«ãç ¥ § ¤ ç¨ „¨à¨å«¥ ¨

X

=

W

1 2() ¢ á«ãç ¥ ¢â®à®© ¨ âà¥â쥩 ªà ¥¢ëå § ¤ ç, á«¥¤ãî騬 ®¡à §®¬: ¤«ï § ¤ ç¨ „¨à¨å«¥ ¨ ¢â®à®© ªà ¥¢®© § ¤ ç¨

J

p(

u

) =

J

0(

u

) +

G

p(

u

)

;

J

0(

u

) = 12 n X i;j=1 Z

a

ij(

x

)

u

xi

u

xj

dx

+ 12 Z

c

(

x

)

u

2(

x

)

dx;

G

p(

u

) = Z

dx

Z u (x) 0

g

(

x;s

)

ds

; Z

p

(

x

)

u

(

x

)

dx;

(1.7) ¢ á«ãç ¥ âà¥â쥣® ªà ¥¢®£® ãá«®¢¨ï (0.3)

J

p(

u

) =

J

0(

u

) + 12 Z ;



(

s

)

u

2(

s

)

ds

+

G

p(

u

)

:

(1.8) ‚ ä®à¬ã«¨à®¢ª å ⥮६ 1.3 ¨ 1.4 ¡ã¤¥¬ ¯®«ì§®¢ âìáï ¢¢¥¤¥­­ë¬¨ ®¡®§­ ç¥­¨ï¬¨ ¡¥§ ¤®¯®«­¨-⥫ì­ëå ¯®ïá­¥­¨©. ’¥®à¥¬  1.3. ãáâì 1) ªà ¥¢ ï § ¤ ç  (0

:

4){(0

:

5) ¨¬¥¥â ­¥­ã«¥¢®¥ à¥è¥­¨¥ ¨

N

(

L

) | ¯®¤¯à®áâà ­á⢮ ¢á¥å ¥¥ à¥è¥­¨©; 2) ¥á«¨

Bu



u

, â®

J

0(

u

) 0 8

u

2 

W

1 2(), ¥á«¨

Bu

 @n@u L, â®

J

0(

u

) 0 8

u

2

W

1 2(), ¥á«¨

Bu

 @n@u L +



(

x

)

u

, â®

J

0(

u

) + 1 2 R ;



(

s

)

u

2(

s

)

ds

0 8

u

2

W

1 2(); 3) ¢ë¯®«­ï¥âáï ãá«®¢¨¥ () ¨ (0

:

6); 4) äã­ªæ¨ï

p

2

L

q() â ª ï, çâ® lim u2N(L);kuk!+1

G

p(

u

) = +1

:

(1.9) ’®£¤  áãé¥áâ¢ã¥â

u

0 2

X

, ¤«ï ª®â®à®£®

J

p(

u

0) = inf X

J

p(

u

)

;

(1.10) ¯à¨ç¥¬ «î¡®¥ â ª®¥

u

0 ¯à¨­ ¤«¥¦¨â

W

2 q(), 㤮¢«¥â¢®àï¥â ¢ª«î祭¨î ;

Lu

0(

x

) +

p

(

x

) 2[

g

;(

x;u

(

x

))

; g

+(

x;u

(

x

))] (1.11)

¤«ï ¯®ç⨠¢á¥å

x

2 ¨ £à ­¨ç­®¬ã ãá«®¢¨î (0

:

2). …᫨ ª ⮬㠦¥ ¤«ï ãà ¢­¥­¨ï (0

:

1) ¢ë-¯®«­¥­®

Ay

(

A

1

y

), â® «î¡®¥

u

0, 㤮¢«¥â¢®àïî饥(1

:

10), ï¥âáï ¯®«ã¯à ¢¨«ì­ë¬ (ᨫì­ë¬) à¥è¥­¨¥¬ § ¤ ç¨ (0

:

1){(0

:

2). ’¥®à¥¬  1.4. ãáâì ¢ë¯®«­¥­ë ãá«®¢¨ï 1) ¨ 3) ⥮६ë 1

:

3,   ¤«ï ä㭪樨

p

2

L

q() «¨¡® ãá«®¢¨¥(1

:

9), «¨¡® lim u2N(L);kuk!+1

G

p(

u

) =;1

:

(1.12) ’®£¤  ¬­®¦¥á⢮ ªà¨â¨ç¥áª¨å §­ ç¥­¨© ä㭪樮­ « 

J

p(

u

) ¢

X

­¥ ¯ãáâ®, ¯à¨ç¥¬ «î¡ ï ªà¨-â¨ç¥áª ï â®çª 

u

0 í⮣® ä㭪樮­ «  ¯à¨­ ¤«¥¦¨â

W

2 q(), 㤮¢«¥â¢®àï¥â ¢ª«î祭¨î (1

:

11) ¨ £à ­¨ç­®¬ã ãá«®¢¨î(0

:

2). …᫨ ª ⮬㠦¥ ¤«ï ãà ¢­¥­¨ï (0

:

1) ¢ë¯®«­¥­® ᨫ쭮¥

Ay

(ᨫ쭮¥

A

1

y

), â® ª ¦¤ ï ªà¨â¨ç¥áª ï â®çª  ä㭪樮­ « 

J

p(

u

) ï¥âáï ¯®«ã¯à ¢¨«ì­ë¬ (ᨫì­ë¬) à¥è¥­¨¥¬ § ¤ ç¨ (0

:

1){(0

:

2). “ª ¦¥¬ ­  á¢ï§ì ãá«®¢¨ï ‹ ­¤¥á¬ ­ {‹ §¥à  [3] á ãá«®¢¨ï¬¨ (1.9), (1.12) ¢ á«ãç ¥, ª®£¤  ¯®¤¯à®áâà ­á⢮

N

(

L

) ®¤­®¬¥à­® ¨ ¤«ï ¯®ç⨠¢á¥å

x

2 áãé¥áâ¢ã¥â lim u!1

g

(

x;u

) =

g

(

x

) (ç¥à¥§ ¡ã¤¥¬ ®¡®§­ ç âì ¡ §¨á­ãî äã­ªæ¨î

N

(

L

)). à¥¤¯®« £ ¥âáï, çâ® äã­ªæ¨ï

g

(

x;u

) á㯥௮-§¨æ¨®­­® ¨§¬¥à¨¬  ¨ ¤«ï ­¥¥ ¢¥à­  ®æ¥­ª  (0.6). “á«®¢¨¥ ‹ ­¤¥á¬ ­ {‹ §¥à  ¨¬¥¥â ¢¨¤: ¤«ï ä㭪樨

p

2

L

q() «¨¡® Z <0

g

+ (

x

)

dx

+ Z >0

g

; (

x

)

dx <

Z

p

(

x

) (

x

)

dx <

Z >0

g

+ (

x

)

dx

+ Z <0

g

; (

x

)

dx;

(1.13) «¨¡® Z <0

g

; (

x

)

dx

+ Z >0

g

+ (

x

)

dx <

Z

p

(

x

) (

x

)

dx <

Z >0

g

; (

x

)

dx

+ Z <0

g

+ (

x

)

dx:

(1.14) à¥¤«®¦¥­¨¥ 1.2. ¥à ¢¥­á⢠ (1

:

13) ((1

:

14)) ¢«¥ªãâ (1

:

9) ((1

:

12)). ‚ [19] ¤®ª § ­® íâ® ã⢥ত¥­¨¥ ¢ á«ãç ¥, ª®£¤ 

g

(

x;u

) ­¥ § ¢¨á¨â ®â

x

. „®ª § â¥«ìá⢮ ¯à¥¤«®¦¥­¨ï 1.2 ®¯¨à ¥âáï ­  á«¥¤ãîéãî «¥¬¬ã. ‹¥¬¬  1.1. ãáâì

f

:

R

!

R

| «®ª «ì­® á㬬¨à㥬 ï äã­ªæ¨ï ¨ áãé¥áâ¢ã¥â lim s!1

f

(

s

) =

f

. ’®£¤  áãé¥áâ¢ã¥â lim !1

f

(

s

)

ds=

=

f

. „®ª § â¥«ìá⢮ «¥¬¬ë 1.1. ®«®¦¨¬
(



) = Z  0

f

(

s

)

ds=

 ;

f

 = Z  0 (

f

(

s

);

f

)

ds=:

®ª ¦¥¬, çâ® lim_ !+1
+(



) = 0. ‚®§ì¬¥¬

" >

0. ®áª®«ìªã lims !+1

f

(

s

) =

f

+, â® áãé¥áâ¢ã¥â



0

>

0 â ª®¥, çâ®j

f

(

s

);

f

+ j

< "=

2 ¤«ï «î¡®£®

s > 

0. „ «¥¥ ¢ë¡¥à¥¬



1

> 

0, ¤«ï ª®â®à®£® 0 R 0 j

f

(

s

);

f

+ j

ds=

1

< "=

2. ’®£¤  ¤«ï ª ¦¤®£®

 > 

1 ¨¬¥¥¬ j
+(



) j  Z  0 0 j

f

(

s

);

f

+ j

ds=

 +Z  0 j

f

(

s

);

f

+ j

ds= < "

2 +



;



0



2

"

< "

2 +

"

2 =

":

‚ ᨫ㠯ந§¢®«ì­®áâ¨

"

§ ª«îç ¥¬, çâ® lim_ !;1
+ = 0. €­ «®£¨ç­® ¤®ª §ë¢ ¥âáï, çâ® lim !;1
;(



) = 0. 

„®ª § â¥«ìá⢮ ¯à¥¤«®¦¥­¨ï 1.2.ãáâì ¢¥à­ë ­¥à ¢¥­á⢠ (1.13) ¨

F

(



) =R

dx

 (x) R 0 (

g

(

x;s

);

p

(

x

))

ds

. „«ï ¯à®¨§¢®«ì­®£®



6= 0 ¨¬¥¥¬

F

(



)

=

=Z (x)6=0

dx

(

x

) 1



(

x

) Z  (x) 0

g

(

x;s

)

ds

;

p

(

x

)  =Z >0  (

x

) 1

_

₍

_x

₎ Z  (x) 0

g

(

x;s

)

ds



dx

+ +Z (x)<0  (

x

) 1

_

₍

_x

₎ Z  (x) 0

g

(

x;s

)

ds



dx

; Z

p

(

x

) (

x

)

dx:

‘®£« á­® «¥¬¬¥ 1.1 ¤«ï ¯®ç⨠¢á¥å

x

2 ¨§ (

x

)

>

0 á«¥¤ã¥â, çâ® lim !1 1  (x)  (x) R 0

g

(

x;s

)

ds

=

g

(

x

),   (

x

)

<

0 ᮮ⢥âá⢥­­® ¢«¥ç¥â lim !1 1  (x)  (x) R 0

g

(

x;s

)

ds

=

g

(

x

). ®í⮬ã, ¥á«¨ ãç¥áâì ­¥à ¢¥­á⢮   1  (x)  (x) R 0

g

(

x;s

)

ds

  

a

(

x

) ¤«ï ¯®ç⨠¢á¥å

x

2, 㤮¢«¥â¢®àïîé¨å ãá«®¢¨î (

x

)6=0, â® ¨§ â¥®à¥¬ë ‹¥¡¥£  ® ¯¥à¥å®¤¥ ª ¯à¥¤¥«ã ¯®¤ §­ ª®¬ ¨­â¥£à «  ¯®«ã稬 lim !1

F

(



)

=

=Z >0 (

x

)

g

(

x

)

dx

+ Z <0 (

x

)

g

(

x

)

dx

; Z

p

(

x

) (

x

)

dx

(

a

(

x

) ¨§ ®æ¥­ª¨ (0.6)). ’®£¤  ¨§ (1.13) á«¥¤ã¥â lim_ !1

F

(



) = +1. €­ «®£¨ç­® à áᬠâਢ ¥âáï á«ãç ©, ª®£¤  ¢ë¯®«­¥­® ãá«®¢¨¥ (1.14)

:

 à¥¤«®¦¥­¨¥ 1.3. ãáâì äã­ªæ¨ï

g

(

x;u

) ¢ ãà ¢­¥­¨¨ (0

:

1) á㯥௮§¨æ¨®­­® ¨§¬¥à¨¬ , ¤«ï ­¥¥ ¢¥à­  ®æ¥­ª (0

:

6) ¨ ¤«ï ¯®ç⨠¢á¥å

x

2 áãé¥áâ¢ã¥â lim u!1

g

(

x;u

) =

g

(

x

),   ¯®¤¯à®-áâà ­á⢮

N

(

L

) ®¤­®¬¥à­®, | ¡ §¨á­ ï äã­ªæ¨ï ¢

N

(

L

), ¨ ¤«ï ¯®ç⨠¢á¥å

x

2

g

;(

x

)

< g

(

x;s

)

< g

+(

x

)(

g

+(

x

)

< g

(

x;s

)

< g

;(

x

)) 8

s

2

R:

(1.15) ’®£¤  ãá«®¢¨¥ (1

:

13) ((1

:

14)) ï¥âáï ­¥®¡å®¤¨¬ë¬ ¤«ï áãé¥á⢮¢ ­¨ï ᨫ쭮£® à¥è¥­¨ï § -¤ ç¨(0

:

1){(0

:

2) ¨§ ¯à®áâà ­á⢠

W

2 q(). „®ª § â¥«ìá⢮ ¯à¥¤«®¦¥­¨ï 1.3. „®¯ãá⨬, çâ®

u

2

W

2 q() | ᨫ쭮¥ à¥è¥­¨¥ § ¤ -ç¨ (0

:

1){(0

:

2). “¬­®¦¨¬ ®¡¥ ç á⨠ãà ¢­¥­¨ï (0.1) ­  ¨ ¯à®¨­â¥£à¨à㥬 ¯®«ã祭­®¥ à ¢¥­-á⢮ ¯® . “ç¨â뢠ï, çâ® R

Lu

(

x

) (

x

)

dx

= R

u

(

x

)

L

(

x

)

dx

= 0, ¯®«ã稬 R

g

(

x;u

(

x

)) (

x

)

dx

= R

p

(

x

) (

x

)

dx

. Žâá ¨ ¨§ ­¥à ¢¥­á⢠ (1.15) á«¥¤ã¥â (1.13) ((1.14))

:

 ‡ ¬¥ç ­¨¥ 1.2. „ ­­®¥ ¤®ª § â¥«ìá⢮ ¯®¢â®àï¥â à áá㦤¥­¨ï ([6], á.54) ¤«ï á«ãç ï, ª®-£¤  ­¥«¨­¥©­®áâì

g

(

x;u

) ª à â¥®¤®à¨¥¢ ,   (0.2) | £à ­¨ç­®¥ ãá«®¢¨¥ „¨à¨å«¥. ®á«¥¤®¢ â¥«ì­®¥ ¯à¨¬¥­¥­¨¥ ¯à¥¤«®¦¥­¨© 1.3 ¨ 1.2 ¤ ¥â ‘«¥¤á⢨¥ 1.1. à¨ ¢ë¯®«­¥­¨¨ ãá«®¢¨© ¯à¥¤«®¦¥­¨ï 1.3 ãá«®¢¨ï (1.9) ((1.12)) ïîâáï ­¥®¡å®¤¨¬ë¬¨ ¤«ï áãé¥á⢮¢ ­¨ï ᨫ쭮£® à¥è¥­¨ï

u

2

W

2 q() § ¤ ç¨ (0.1){(0.2). 2. „®ª § â¥«ìá⢮ ®¡é¨å ¢ à¨ æ¨®­­ëå ¯à¨­æ¨¯®¢ „®ª § â¥«ìá⢮ ⥮६ë 1.1. ’ ª ª ª ­ã«ì | ¨§®«¨à®¢ ­­ ï â®çª  ᯥªâà  á ¬®á®¯àï-¦¥­­®£® ®¯¥à â®à 

A

, â®

X

à §« £ ¥âáï ¢ ¯àï¬ãî á㬬㠮à⮣®­ «ì­ëå ¯®¤¯à®áâà ­áâ¢

N

(

A

) ¨

X

+ (

X

+ | ¯®«®¦¨â¥«ì­®¥ ¯®¤¯à®áâà ­á⢮ ®¯¥à â®à 

A

) ¨ áãé¥áâ¢ã¥â ç¨á«®

 >

0 â ª®¥, çâ® (

Ax;x

) 



k

x

k 2 8

x

2

X

+, £¤¥ k
k | ­®à¬  ¢ £¨«ì¡¥à⮢®¬ ¯à®áâà ­á⢥

X

. ˆ§ á ¬®á®¯àï-¦¥­­®á⨠®¯¥à â®à 

A

á«¥¤ã¥â, çâ® (

Ax;x

)

=

2 | ¥£® ¯®â¥­æ¨ «. Žâá ¨ ¨§ ª¢ §¨¯®â¥­æ¨ «ì­®-á⨠®¯¥à â®à 

T

á«¥¤ã¥â ª¢ §¨¯®â¥­æ¨ «ì­®áâì ®â®¡à ¦¥­¨ï (1.1) ¨ ã⢥ত¥­¨¥: ä㭪樮­ «

'

(

x

) = (

Ax;x

)

=

2 +

f

(

x

);(

p;x

) | ª¢ §¨¯®â¥­æ¨ «

Q

.

‚ ᨫã ãá«®¢¨ï 2) ⥮६ë 1.1 ®¯¥à â®à 

A

¬®­®â®­­ë© ([13], c.22) ¨ ­¥¯à¥à뢭ë©. ®ª -¦¥¬, çâ® lim kxk!+1

'

(

x

) = +1

:

(2.1) „«ï «î¡®£®

x

2

X

,

x

=

x

1 +

x

2

;x

1 2

N

(

A

),

x

2 2

X

+, ¨¬¥¥¬

'

(

x

) = (

Ax

2

;x

2)

=

2 + (

f

(

x

1 +

x

2) ;

f

(

x

1))+(

f

(

x

1) ;(

p;x

))  2 k

x

2 k 2 ;(

M

1+ k

p

k)k

x

2 k+(

f

(

x

1) ;(

p;x

1)), £¤¥

M

1 =

M

k

P

k,

M

| ¯®áâ®ï­­ ï ¨§ ãá«®¢¨ï 3 ⥮६ë 1.1 (¢ ᨫ㠧 ¬¥ç ­¨ï 1.1 j

f

(

x

);

f

(

y

)j

M

k

x

;

y

kY 

M

1 k

x

;

y

k 8

x;y

2

X

). ”¨ªá¨à㥬

" >

0. ‚ ᨫã (1.3) ­ ©¤¥âáï

d

0

>

0 â ª®¥, çâ®

f

(

x

) ;(

p;x

) 0 ¤«ï «î¡®£®

x

2

N

(

A

) á k

x

k 

d

0, ¨ áãé¥áâ¢ã¥â

d

1

>

0, ¤«ï ª®â®à®£® ¨§ k

x

k 

d

1,

x

2

N

(

A

) á«¥¤ã¥â

f

(

x

);(

p;x

)

> "

+ (M1+kpk) 2 2 . ãáâì

d

2

> d

0 ¨ ¤«ï ­¥£® ­¥à ¢¥­á⢮

t > d

2 ¢«¥ç¥â



2

t

2 ;(

M

1+ k

p

k)

t > "

;min  0

;

_x inf 2N(A);kxkd 0 (

f

(

x

);(

p;x

))

:

ˆ­ä¨­ã¬ ¢ ¯à ¢®© ç á⨠¯®á«¥¤­¥£® ­¥à ¢¥­á⢠ ª®­¥ç­ë©, ¯®áª®«ìªã j

f

(

x

)jj

f

(

x

);

f

(0)j+ j

f

(0)j 

M

1 k

x

k+j

f

(0)j 8

x

2

N

(

A

). ’ ª ª ª k

x

k = (k

x

1 k 2 + k

x

2 k 2)1=2,

x

=

x

1+

x

2

;x

1 2

N

(

A

),

x

2 2

X

+, â® ¨§ ­¥à ¢¥­á⢠ k

x

k

>

p 2maxf

d

1

;d

2 g á«¥¤ã¥â, çâ® «¨¡® k

x

1 k

> d

1, «¨¡® k

x

2 k

> d

2. ‚ ¯¥à¢®¬ á«ãç ¥

'

(

x

)

>

; (M1+kpk) 2 2 +

"

+ (M1+kpk) 2 2 =

"

, â.ª.  2

t

2 ;(

M

1+ k

p

k)

t

 (M1+kpk) 2 2 8

t

2

R

; ¢® ¢â®à®¬ á«ãç ¥

'

(

x

)

> "

;min  0

;

_x inf 2N(A);kxkd 0 (

f

(

x

);(

p;x

)) +

f

(

x

1) ;(

p;x

1) 

"

. ’ ª¨¬ ®¡à §®¬, ¤«ï «î¡®£®

x

2

X

á k

x

k

>

p 2maxf

d

1

;d

2 g

'

(

x

)

> "

. ®í⮬㠢 ᨫ㠯ந§¢®«ì­®á⨠¢ë¡®à 

" >

0 § ª«îç ¥¬ ® á¯à ¢¥¤«¨¢®á⨠(2.1). ‚믮«­¥­ë ¢á¥ ãá«®¢¨ï ¢ à¨ æ¨®­­®£® ¯à¨­æ¨¯  ([20], á.16). ‘«¥¤®¢ â¥«ì­®, áãé¥áâ¢ã¥â

x

0 2

X

, ¤«ï ª®â®à®£®

'

(

x

0) = inf X

'

(

x

)

;

(2.2) ¯à¨ç¥¬ «î¡®¥ â ª®¥

x

0㤮¢«¥â¢®àï¥â ¢ª«î祭¨î (1.4). …᫨ ¤®¯®«­¨â¥«ì­® ¨§¢¥áâ­®, çâ® â®çª¨ à §à뢠 ®¯¥à â®à 

Q

ॣã«ïà­ë, â® ¢á类¥

x

0 2

X

, ¤«ï ª®â®à®£® ¢¥à­® (2.2), 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î (1.1) ¨ ï¥âáï â®çª®© à ¤¨ «ì­®© ­¥¯à¥à뢭®á⨠®¯¥à â®à 

Q

([20], c.13)

:

 „®ª § â¥«ìá⢮ ⥮६ë 1.2. „®ª § â¥«ìá⢮ ­¥¯ãáâ®âë ¬­®¦¥á⢠ ªà¨â¨ç¥áª¨å â®ç¥ª ä㭪樮­ « 

'

(

x

) = (

Ax;x

)

=

2 +

f

(

x

);(

p;x

) ®¯¨à ¥âáï ­  á«¥¤ãî騩 १ã«ìâ â —¥­£ . ’¥®à¥¬  2.1 ([9]). ãáâì

E

| ¢¥é¥á⢥­­®¥ à¥ä«¥ªá¨¢­®¥ ¡ ­ å®¢® ¯à®áâà ­á⢮, äã­ª-æ¨ï

'

:

E

!

R

«®ª «ì­® «¨¯è¨æ¥¢  ¨ 㤮¢«¥â¢®àï¥â (P

:

S

:

) ãá«®¢¨î. à¥¤¯®«®¦¨¬, çâ®

E

=

E

1 

E

2, £¤¥ ¯®¤¯à®áâà ­á⢮

E

1 ª®­¥ç­®¬¥à­®, ¨ áãé¥áâ¢ãîâ ¯®áâ®ï­­ë¥

b

1

< b

2 ¨ ®ªà¥áâ­®áâì

N

­ã«ï ¯à®áâà ­á⢠

E

1 â ª¨¥, çâ®

'

jE 2 

b

2

;'

j@N 

b

1 (

@N

| £à ­¨æ 

N

). ’®£¤  ¬­®¦¥á⢮ ªà¨â¨ç¥áª¨å â®ç¥ª

'

­¥¯ãáâ®. ƒ®¢®àïâ, çâ® «®ª «ì­® «¨¯è¨æ¥¢  äã­ªæ¨ï

'

:

E

!

R

㤮¢«¥â¢®àï¥â (P.S.) ãá«®¢¨î [9], ¥á«¨ «î¡ ï ¯®á«¥¤®¢ â¥«ì­®áâì (

x

n)

E

, ¤«ï ª®â®à®© ¬­®¦¥á⢮ §­ ç¥­¨© (

'

(

x

n)) ®£à ­¨ç¥­® ¨



(

x

n) = min_w 2@'(xn) k

w

kE !0, ᮤ¥à¦¨â á室ïéãîáï ¯®¤¯®á«¥¤®¢ â¥«ì­®áâì. ”ã­ªæ¨ï

'

(

x

) = (

Ax;x

)

=

2+

f

(

x

);(

p;x

) «®ª «ì­® «¨¯è¨æ¥¢  ­  X, ¯®áª®«ìªã

A

| «¨­¥©­ë© ®£à ­¨ç¥­­ë© ®¯¥à â®à,  

f

«¨¯è¨æ¥¢  ­ 

Y

(á¬. § ¬¥ç ­¨¥ 1.1). ® ãá«®¢¨î ¯à®áâà ­á⢮

X

ª®¬¯ ªâ­® ¨ ¯«®â­® ¢«®¦¥­® ¢

Y

, ®¯¥à â®à

A

á ¬®á®¯à殮­­ë©, ­ã«ì | ¨§®«¨à®¢ ­­ ï â®çª  ¥£® ᯥªâà , ¯à¨ç¥¬ ï¤à®

N

(

A

) ¨ ®âà¨æ â¥«ì­®¥ ¯®¤¯à®áâà ­á⢮ ®¯¥à â®à 

A

ª®­¥ç­®¬¥à­ë ¨ ¤«ï

p

¢¥à­® (1.5). Žâá á«¥¤ã¥â, çâ®

'

㤮¢«¥â¢®àï¥â (P.S.) ãá«®¢¨î ([9], ⥮६  4.5). à®¢¥à¨¬, çâ® ¤«ï

'

¢ë¯®«­¥­ë ¨ ®áâ «ì­ë¥ ãá«®¢¨ï ⥮६ë 2.1, á¢ï§ ­­ë¥ á à §«®¦¥­¨¥¬ ¯à®áâà ­á⢠, ­  ª®â®à®¬ ®¯à¥¤¥«¥­ 

'

, ¢ ¯àï¬ãî á㬬ã. ˆ§ ãá«®¢¨ï 2) ⥮६ë 1.2 á«¥¤ã¥â, çâ®

X

à á¯ ¤ ¥âáï ­  ¯àï¬ãî á㬬㠮à⮣®­ «ì­ëå ¨ ¨­¢ à¨ ­â­ëå ®â­®á¨â¥«ì­®

A

¯®¤¯à®áâà ­áâ¢

N

(

A

),

X

; ¨

X

+ ¨ áãé¥áâ¢ãîâ ¯®«®¦¨â¥«ì­ë¥ ª®­áâ ­âë



¨



â ª¨¥, çâ® (

Ax;x

) 



k

x

k 2 8

x

2

X

+

;

(

Ax;x

) ;



k

x

k 2 8

x

2

X

;

:

ãáâì ¤«ï ®¯à¥¤¥«¥­­®áâ¨_x lim 2N(A);kxk!+1 (

f

(

x

);(

p;x

)) = +1. ®«®¦¨¬

X

1=

X

;,

X

2 =

N

(

A

) 

X

+, ⮣¤ 

X

=

X

1 

X

2 ¨

X

1 ª®­¥ç­®¬¥à­®¥. „«ï ¯à®¨§¢®«ì­®£®

x

2

X

2,

x

=

x

0+

x

1,

x

0 2

N

(

A

),

x

1 2

X

+ ¨¬¥¥¬ á ãç¥â®¬ § ¬¥ç ­¨ï 1.1

'

(

x

) = (

Ax;x

)

=

2 + (

f

(

x

0+

x

1) ;

f

(

x

0)) + (

f

(

x

0) ;(

p;x

0)) ;(

p;x

1)  



2k

x

1 k 2 ;(

M

1+ k

p

k)k

x

1 k+ (

f

(

x

0) ;(

p;x

0)) ; (

M

1+ k

p

k) 2 2



+ infv2N(A) (

f

(

v

);(

p;v

)) =

b;

£¤¥

M

1=

M

k

P

k,

M

| ¯®áâ®ï­­ ï ¢ ãá«®¢¨¨ 3) ⥮६ë 1.1. „ «¥¥, ¤«ï

x

2

X

1

'

(

x

) = (

Ax;x

)

=

2 +

f

(

x

);

f

(0) +

f

(0);(

p;x

);



2k

x

k 2+ (

M

1+ k

p

k)k

x

k+j

f

(0)j

:

®í⮬ã áãé¥áâ¢ã¥â â ª®¥

r >

0, çâ®

b

1 = sup x2X; kxk=r

'

(

x

)

< b

2, ¨, §­ ç¨â, ¢á¥ ãá«®¢¨ï ⥮६ë 2.1 ¢ë¯®«­¥­ë. ‚ á«ãç ¥, ª®£¤  lim x2N(A);kxk!+1 (

f

(

x

);(

p;x

)) =;1

;

(2.3) ¯®« £ ¥¬

X

1 =

X

; 

N

(

A

),

X

2 =

X

+. ’®£¤ 

X

=

X

1 

X

2 ¨

X

1 ª®­¥ç­®¬¥à­®¥. „«ï ª ¦¤®£®

x

2

X

2 ¨¬¥¥¬

'

(

x

) = (

Ax;x

)

=

2 + (

f

(

x

);

f

(0)) + (

f

(0);(

p;x

))



2k

x

k 2 ; ;(

M

1+ k

p

k)k

x

k;j

f

(0)j; (

M

1+ k

p

k) 2 2



;j

f

(0)j=

b

2

:

Žæ¥­¨¬

'

ᢥàåã ­ 

X

1. „«ï «î¡®£®

x

2

X

1,

x

=

x

0+

x

2,

x

0 2

N

(

A

),

x

2 2

X

;,

'

(

x

) = (

Ax;x

)

=

2 + (

f

(

x

0+

x

2) ;

f

(

x

0)) + (

f

(

x

0) ;(

p;x

0)) ;(

p;x

2)  ;



2k

x

2 k 2 ;(

M

1+ k

p

k)k

x

2 k+ (

f

(

x

0) ;(

p;x

0))

:

‚ ᨫã (2.3) áãé¥áâ¢ã¥â

d

1

>

0 â ª®¥, çâ® ¤«ï ¯à®¨§¢®«ì­®£®

x

0 2

N

(

A

) c k

x

0 k

d

1

f

(

x

0) ; (

p;X

0)

< b

2 ; (M 1 +kpk) 2 2 .  ©¤¥âáï

d

2

>

0, ¤«ï ª®â®à®£® ­¥à ¢¥­á⢮ k

x

2 k

d

2,

x

2 2

X

; ¢«¥ç¥â ;



2k

X

2 k 2 + (

M

1+ k

p

k)k

x

2 k

< b

2 ; sup 2N(A) (

f

(



);(

p;

))

:

’®£¤ , ¥á«¨

x

2

X

1,

x

=

x

0+

x

2,

x

0 2

N

(

A

)

;x

2 2

X

;, ¨ k

x

k= p k

x

0 k 2+ k

x

2 k 2  p 2max

d

1,

d

2 =

r

, «¨¡®k

x

0 k

d

1, «¨¡® k

x

2 k

d

2. ‚ ¯¥à¢®¬ á«ãç ¥

'

(

x

)

<

(

M

1+ k

p

k) 2 2



+

b

2 ; (

M

1+ k

p

k) 2 2



=

b

2; ¢® ¢â®à®¬

'

(

x

)

< b

2 ; sup 2N(A) (

f

(



);(

p;

)) +

f

(

x

0) ;(

p;x

0) 

b

2. ‘«¥¤®¢ â¥«ì­®,

b

1 = sup x2X 1; kxk=r

'

(

x

)

< b

2, ¯®áª®«ìªã áä¥à  k

x

k =

r

¢ ª®­¥ç­®¬¥à­®¬ ¯à®-áâà ­á⢥

X

1 ª®¬¯ ªâ­ ,   á㦥­¨¥

'

jX 1 ­¥¯à¥à뢭® ­  ­¥©. ’ ª¨¬ ®¡à §®¬, ¨ ¢ í⮬ á«ãç ¥ ¢á¥ ãá«®¢¨ï ⥮६ë 2.1 ¢ë¯®«­¥­ë. ‘«¥¤®¢ â¥«ì­®, áãé¥áâ¢ã¥â

x

0 2

X

â ª ï, çâ® 0 2

@'

(

x

0),

@'

(

x

0) | ®¡®¡é¥­­ë© £à ¤¨¥­â Š« àª  ä㭪樨

'

. ˆ§ ®¯à¥¤¥«¥­¨ï

@'

(

x

0) § ª«îç ¥¬, çâ® ¤«ï ¯à®¨§¢®«ì­®£®



2

X

;(

Ax

0

;

) + (

p;

)  limsup h2X; h!0 t!+0

f

(

x

0+

h

+

t

) ;

f

(

x

0+

h

)

t

:

(2.4)

’ ª ª ª

f

jX | ª¢ §¨¯®â¥­æ¨ « ®¯¥à â®à 

P



TP

, â®

f

(

x

0+

h

+

t

) ;

f

(

x

0+

h

)

t

= Z 1 0 h

P



TP

(

x

0+

h

+

t

)

;

i

d:

®í⮬㠨§ (2.4) ¯®«ã稬 limsup h2X; h!0 t!+0 Z 1 0 h

P



TP

(

x

0+

h

+

t

)

;

i

d

+ (

Ax

0

;

) ;(

p;

)0 8



2

X:

(2.5) ®ª ¦¥¬, çâ® (2.5) ¢«¥ç¥â

y

0= ;

Ax

0+ 

p

2

S

(

P



TP

)(

x

0)

;

(2.6) £¤¥

S

(

P



TP

) | ᥪ¢¥­æ¨ «ì­®¥ § ¬ëª ­¨¥ ®¯¥à â®à 

F

=

P



TP

[17]. ® ®¯à¥¤¥«¥­¨î

SF

(

x

0) | § ¬ª­ãâ ï ¢ë¯ãª« ï ®¡®«®çª  ¬­®¦¥á⢠ ¢á¥å á« ¡® ¯à¥¤¥«ì­ëå â®ç¥ª ¯®á«¥¤®¢ â¥«ì­®á⥩ ¢¨¤  (

F

(

x

n)) ¢

X

, £¤¥ (

x

n) ᨫ쭮 á室¨âáï ª

x

0. „®¯ãá⨬, çâ® (2.6) ­¥ ¨¬¥¥â ¬¥áâ . ‘®£« á­® ⥮६¥ ® áâண®© ®â¤¥«¨¬®á⨠¢ë¯ãª«®£® ¬­®¦¥á⢠ ®â â®çª¨, áãé¥áâ¢ãîâ



0 2

X

¨

" >

0 â ª¨¥, çâ® h

z;

0 i;h

y

0

;

0 i

<

;

"

8

z

2

SF

(

x

0)

:

Žâá á«¥¤ã¥â, çâ® limsup h2X; h!0 t!+0 h

F

(

x

0+

h

+

s

0)

;

0 i;h

y

0

;

0 i

<

;

":

(2.7) ˆ§ (2.7) § ª«îç ¥¬ ® áãé¥á⢮¢ ­¨¨

 >

0, ¤«ï ª®â®à®£® (

F

(

x

0 +

h

+

s

0)

;

0) ;(

y

0

;

0)

<

;

"

, ¥á«¨k

h

k

< 

¨ 0

< s < 

. ®í⮬㠯®«ã稬 limsup h2X; h!0 t!+0 Z 1 0 h

F

(

x

0+

h

+

t

0)

;

0 i

d

;h

y

0

;

0 i;

";

çâ® ¯à®â¨¢®à¥ç¨â (2.5) ¯à¨



=



0. ’ ª¨¬ ®¡à §®¬, ¤®ª § ­  ­¥¯ãáâ®â  ¬­®¦¥á⢠ ªà¨â¨ç¥áª¨å â®ç¥ª ä㭪樨

'

(

x

) ¨ â®, çâ® «î¡ ï â ª ï â®çª 

x

0 㤮¢«¥â¢®àï¥â ¢ª«î祭¨î (2.6). Š ª ¯®ª § ­® ¢ [21],

S

(

P



TP

)

x

0 

P

(

ST

)(

Px

0) ([20], c.16), ¯®í⮬ã (2.6) ¢«¥ç¥â (1.4). …᫨ ¤®¯®«­¨â¥«ì­® ¯à¥¤¯®«®¦¨âì, çâ® ¢á¥ â®çª¨ à §à뢠 ®¯¥à â®à  (1.1) ᨫ쭮 ॣã«ïà­ë¥, â® «î¡ ï ªà¨â¨ç¥áª ï â®çª 

x

0 äã­ª-樨

'

ï¥âáï â®çª®© ¤¥¬¨­¥¯à¥à뢭®á⨠®¯¥à â®à 

P



TP

¨ 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î (1.1). „¥©á⢨⥫쭮, ¥á«¨ ¤®¯ãáâ¨âì, çâ® ªà¨â¨ç¥áª ï â®çª 

x

0 ä㭪樮­ « 

'

¥áâì â®çª  à §à뢠 ¤«ï

P



TP

, â® ®­  ¤®«¦­  ¡ëâì ᨫ쭮 ॣã«ïà­®© ¤«ï

Q

, â.¥. áãé¥áâ¢ã¥â



0 2

X

, ¤«ï ª®â®à®© limsup_h 2X h!0 h

Q

(

x

0+

h

)

;

0 i

<

0. Žâá ¨ ¨§ ®¯à¥¤¥«¥­¨ï ¢¥àå­¥£® ¯à¥¤¥«  á«¥¤ã¥â áãé¥á⢮¢ ­¨¥ ¯®«®¦¨â¥«ì­ëå ç¨á¥«

s

¨

"

â ª¨å, çâ® ¤«ï «î¡ëå

h

2

X

, k

h

k

< "

¨¬¥¥¬ h

Q

(

x

0+

h

)

;

0 i

<

;

s

. à¨ í⮬, ¥á«¨

h

2

X

,

t >

0 ¨k

h

k+

t

k



0 k

< "

, â® h

Q

(

x

0+

h

+

t

0)

;

0 i

<

;

s

8



2[0

;

1]. ®í⮬㠫¥¢ ï ç áâì ­¥à ¢¥­á⢠ (2.5) ¯à¨



=



0 ¬¥­ìè¥ ¨«¨ à ¢­  ;

s

. ‘ ¤à㣮© áâ®à®­ë, ¯®áª®«ìªã

x

0 | ªà¨â¨ç¥áª ï â®çª  ä㭪樨

'

, â® ¤«ï ­¥¥ ¢¥à­® ­¥à ¢¥­á⢮ (2.5) ¤«ï «î¡®£®



2

X

. ®-«ã祭­®¥ ¯à®â¨¢®à¥ç¨¥ ¤®ª §ë¢ ¥â, çâ®

x

0 | â®çª  ¤¥¬¨­¥¯à¥à뢭®á⨠®¯¥à â®à 

Q

. Žâá ¨ ¨§ (2.5), ¨á¯®«ì§ãï ⥮६㠋¥¡¥£  ® ¯¥à¥å®¤¥ ª ¯à¥¤¥«ã ¯®¤ §­ ª®¬ ¨­â¥£à « , ¯®«ã稬 h

Q

(

x

0)

;

i0 8



2

X:

®á«¥¤­¥¥ ¢®§¬®¦­® ⮫쪮 ¯à¨

Q

(

x

0) = 0

:



í««¨¯â¨ç¥áª®£® ⨯  á à §à뢭묨 ­¥«¨­¥©­®áâﬨ ’ ª ª ª ¤¨ää¥à¥­æ¨ «ì­ë© ®¯¥à â®à

L

¢ ãà ¢­¥­¨¨ (0.1) à ¢­®¬¥à­® í««¨¯â¨ç¥áª¨© ¨ ᨬ-¬¥âà¨ç­ë©,   ¥£® ª®íää¨æ¨¥­âë

a

ij 2

C

1;(), â® à ¢¥­á⢠ (

u;

)0= n X i;j=1 Z

a

ij(

x

)

u

xi



xj

dx

8

u;

2 

W

1 2()

;

(

u;

)1= (

u;

)0+ Z

u dx

8

u;

2

W

1 2() § ¤ îâ ­  ¯à®áâà ­á⢠å 

W

1 2() ¨

W

1 2() ᮮ⢥âá⢥­­® ᪠«ïà­ë¥ ¯à®¨§¢¥¤¥­¨ï, ¯à¨ç¥¬ ¯®-஦¤ ¥¬ë¥ ¨¬¨ ­®à¬ë íª¢¨¢ «¥­â­ë ­®à¬ ¬ íâ¨å ¯à®áâà ­áâ¢. —¥à¥§

X

®¡®§­ ç ¥¬ 

W

1 2() ¢ á«ãç ¥ § ¤ ç¨ „¨à¨å«¥ ¨

W

1 2(), ¥á«¨ à áᬠâਢ ¥âáï § ¤ ç  ¥©¬ ­  ¨«¨ âà¥âìï ªà ¥¢ ï § ¤ ç . ‹¨­¥©­ë© ®¯¥à â®à

A

:

X

!

X

®¯à¥¤¥«ï¥âáï à ¢¥­á⢮¬ (

Au;

) = (

u;

)0+ Z

c

(

x

)

u

(

x

)



(

x

)

dx

8

u;

2

X;

¤«ï § ¤ ç „¨à¨å«¥ ¨ ¥©¬ ­ , ¨ (

Au;

) = (

u;

)1+ Z (

c

(

x

);1)

u

(

x

)



(

x

)

dx

+ Z ;



(

x

)

u

(

x

)



(

x

)

dx

8

u;

2

X;

£¤¥ (

;

) | ᪠«ïà­®¥ ¯à®¨§¢¥¤¥­¨¥ ¢

X

,

c

2

C

0;() | ª®íää¨æ¨¥­â ¯à¨

u

(

x

) ¢ ãà ¢­¥­¨¨ (0.1). ‡ ¬¥â¨¬, çâ® ®¯¥à â®à

A

á ¬®á®¯à殮­­ë© ¨ à ¢¥­ á㬬¥ ⮦¤¥á⢥­­®£® ¨ ª®¬¯ ªâ­®£® ®¯¥à -â®à®¢, ¥£® ï¤à® ᮢ¯ ¤ ¥â á ¯®¤¯à®áâà ­á⢮¬

N

(

L

) à¥è¥­¨© § ¤ ç¨ (0.4){(0.5). ‘®£« á­® ⥮ਨ ”।£®«ì¬  ®âà¨æ â¥«ì­®¥ ¯®¤¯à®áâà ­á⢮

X

; ®¯¥à â®à 

A

ª®­¥ç­®¬¥à­® ¨, ¥á«¨

N

(

L

) 6 =f0g, â® ­ã«ì | ¨§®«¨à®¢ ­­ ï â®çª  ᯥªâà  ®¯¥à â®à 

A

ª®­¥ç­®© ªà â­®á⨠([22], c.101). à®-áâà ­á⢮

X

¯«®â­® ¨ ª®¬¯ ªâ­® ¢«®¦¥­® ¢ à¥ä«¥ªá¨¢­®¥ ¡ ­ å®¢® ¯à®áâà ­á⢮

Y

=

L

p(),

p

= q q;1 (

q >

2

n=

(

n

+ 2) ¨§ ®æ¥­ª¨ (0.6)). ‚ ᨫã ãá«®¢¨ï 3) ⥮६ë 1.3 ®¯¥à â®à ¥¬ë檮£®

Tu

=

g

(

x;u

(

x

)), ¯®à®¦¤ ¥¬ë© ­¥«¨­¥©­®áâìî

g

(

x;u

) ¨§ ãà ¢­¥­¨ï (0.1), ¤¥©áâ¢ã¥â ¨§

Y

¢

Y

 ¨ ®£à ­¨ç¥­ ­ 

Y

. Ž­ ª¢ §¨¯®â¥­æ¨ «ì­ë© ­ 

Y

, ¨ ¥£® ª¢ §¨¯®â¥­æ¨ «

f

(

u

) =R

dx

u(x) R 0

g

(

x;s

)

ds

8

u

2

Y

[23]. „®ª § â¥«ìá⢮ ⥮६ë 1.3ˆ§ ãá«®¢¨ï 2) ⥮६ë 1.3 á«¥¤ã¥â, çâ® (

Au;u

)08

u

2

X

. Ž£à ­¨ç¥­¨¥ (1.9) ­  í«¥¬¥­â

p

2

Y

 ¢«¥ç¥â à ¢¥­á⢮ (1.3). ®í⮬㠨§ ⥮६ë 1.1 § ª«î-ç ¥¬ ® áãé¥á⢮¢ ­¨¨

u

0 2

X

, ¤«ï ª®â®à®£®

'

(

u

0) = inf X

'

(

u

),

'

(

u

) = (

Au;u

)

=

2 +

f

(

u

);(

p;u

), ¯à¨ç¥¬ ª ¦¤®¥ â ª®¥

u

0 㤮¢«¥â¢®àï¥â ¢ª«î祭¨î (1.4), £¤¥  | «¨­¥©­ë© ¨§®¬®à䨧¬, ®â®-¦¤¥á⢫ïî騩

X

á

X

. ®á«¥¤­¥¥ ®§­ ç ¥â, çâ® áãé¥áâ¢ã¥â

z

2

STu

0 

L

q(), ¤«ï ª®â®à®£® ¢¥à­® ⮦¤¥á⢮ (

Au

0

;

) = Z (

p

(

x

);

z

(

x

))



(

x

)

dx

8



2

X;

â.¥.

u

0 | ®¡®¡é¥­­®¥ à¥è¥­¨¥ § ¤ ç¨

Lu

(

x

) =

p

(

x

);

z

(

x

)

; x

2

;

(3.1)

Bu

j ; = 0

:

(3.2) à¨ ᤥ« ­­ëå ¢ëè¥ ¯à¥¤¯®«®¦¥­¨ïå ®â­®á¨â¥«ì­® ª®íää¨æ¨¥­â®¢ ¤¨ää¥à¥­æ¨ «ì­®£® ®¯¥à â®à 

L

, £« ¤ª®á⨠£à ­¨æë ; ®¡« á⨠, ä㭪樨



¢ £à ­¨ç­®¬ ãá«®¢¨¨ (0.3) ¨ ¯à¨­ ¤-«¥¦­®áâ¨

p

(

x

);

z

(

x

) ª

L

q() á

q >

2

n=

(

n

+2), ®¡®¡é¥­­®¥ à¥è¥­¨¥ § ¤ ç¨ (3.1){(3.2) ¯à¨­ ¤«¥-¦¨â

W

q 2() ¨ ï¥âáï ᨫì­ë¬ à¥è¥­¨¥¬ í⮩ § ¤ ç¨. „®ª § â¥«ìá⢮ í⮣® ä ªâ  ¢ á«ãç ¥

q

= 2 ¬®¦­® ­ ©â¨ ¢ [1], â ¬ ¦¥ ¯à¨¢®¤¨âáï á奬  ¤®ª § â¥«ìá⢠ ¨ ¯à¨

q

6= 2. ’ ª ª ª

Y

|

à¥ä«¥ªá¨¢­®¥ ¡ ­ å®¢® ¯à®áâà ­á⢮,   ®¯¥à â®à

T

:

Y

!

Y

 «®ª «ì­® ®£à ­¨ç¥­­ë© ­ 

Y

, â®

ST

=

T

}, £¤¥

T

} | ®¢ë¯ãª«¥­¨¥ ®¯¥à â®à 

T

[24]. „«ï ®¯¥à â®à®¢ ¥¬ë檮£®, ¤¥©áâ¢ãîé¨å ¢ «¥¡¥£®¢ëå ¯à®áâà ­á⢠å, ®¢ë¯ãª«¥­¨ï ¡ë«¨ ®¯¨á ­ë Œ.€.Šà á­®á¥«ì᪨¬ ¨ €.‚.®ªà®¢áª¨¬ ¢ ([2], £«.5, x27). ‚ à áᬠâਢ ¥¬®¬ á«ãç ¥ ¤«ï «î¡®£®

u

2

L

p(),

p

=

q=

(

q

;1),

T

}

u

= f

z

: 2

R

j

z

| ¨§¬¥à¨¬ ï ¯® ‹¥¡¥£ã ­ 

;

z

(

x

)2[

g

;(

x;u

(

x

))

;g

+(

x;u

(

x

))] ¤«ï ¯.¢.

x

2g

:

‘«¥¤®¢ â¥«ì­®, ᨫ쭮¥ à¥è¥­¨¥

u

0 § ¤ ç¨ (3.1){(3.2) 㤮¢«¥â¢®àï¥â ¢ª«î祭¨î (1.11). ’ -ª¨¬ ®¡à §®¬, ¤®ª § ­®, çâ® «î¡®¥

u

0 2

X

, ¤«ï ª®â®à®£®

'

(

u

0) = inf X

'

(

u

), 㤮¢«¥â¢®àï¥â ¢ª«î-祭¨î (1.11). à¥¤¯®«®¦¨¬ ⥯¥àì ¤®¯®«­¨â¥«ì­®, çâ® ¤«ï ãà ¢­¥­¨ï (0.1) ¢ë¯®«­¥­®

Ay

. „®ª ¦¥¬, çâ® íâ® ¢«¥ç¥â ॣã«ïà­®áâì â®ç¥ª à §à뢠 ®¯¥à â®à  (1.1). …᫨ ¤«ï ä㭪樨

u

2

X

¬¥à  ¬­®-¦¥á⢠ (

u

) = f

x

2 j

u

(

x

) | â®çª  à §à뢠

g

(

x;:

)g à ¢­  ­ã«î, â®

u

| â®çª  à ¤¨ «ì­®© ­¥¯à¥à뢭®á⨠®¯¥à â®à 

Q

. „¥©á⢨⥫쭮, ¢ í⮬ á«ãç ¥ ¤«ï «î¡®£®

h

2

X

áãé¥áâ¢ã¥â lim t!0 h

Q

(

u

+

th

)

;h

i= lim t!0 f(

A

(

u

+

th

)

;h

) + Z

g

(

x;u

(

x

) +

th

(

x

))

h

(

x

)

dx

; Z

p

(

x

)

h

(

x

)

dx

g= = (

Au;h

); Z

p

(

x

)

h

(

x

)

dx

+Z lim t!0

g

(

x;u

(

x

) +

th

(

x

))

h

(

x

)

dx

= = (

Au;h

); Z

p

(

x

)

h

(

x

)

dx

+Z

g

(

x;u

(

x

))

h

(

x

)

dx

=h

Qu;h

i (¢®á¯®«ì§®¢ «¨áì ⥮६®© ‹¥¡¥£  ® ¯¥à¥å®¤¥ ª ¯à¥¤¥«ã ¯®¤ §­ ª®¬ ¨­â¥£à «  ¨ ®æ¥­ª®© (0.6)). ‘«¥¤®¢ â¥«ì­®, ¥á«¨

u

| â®çª  à §à뢠 ®¯¥à â®à 

Q

, â® mes(

u

)6= 0. „«ï

v

2

X

¬­®¦¥á⢮ (

v

) á â®ç­®áâìî ¤® ¬­®¦¥á⢠ ¬¥àë ­ã«ì ᮢ¯ ¤ ¥â á ®¡ê¥¤¨­¥­¨-¥¬ ¬­®¦¥á⢠+(

v

) = f

x

2 j

g

(

x;v

(

x

)+)

> g

(

x;v

(

x

);)g ¨ ;(

v

) = f

x

2 j

g

(

x;v

(

x

)+)

<

g

(

x;v

(

x

);)g. ãáâì

v

2

X

| â®çª  à §à뢠 ®¯¥à â®à 

Q

. ɇǬ mes +(

v

) = 0, â®

g

(

x;v

(

x

)+) 

g

(

x;v

(

x

);) ¯®ç⨠¢áî¤ã ­  , ¯®í⮬ã lim_t !+0 h

T

(

v

+

th

)

;h

i h

Tv;h

i 8

h

2

X

. Žâá ¨ ¨§ ¯à¥¤«®¦¥­¨ï 1.1 á«¥¤ã¥â, çâ®

v

| ॣã«ïà­ ï â®çª  ¤«ï

Q

. ãáâì ⥯¥àì mes+(

v

) 6 = 0. à¥¤¯®«®¦¨¬, çâ®

v

­¥ ï¥âáï ॣã«ïà­®© ¤«ï ®¯¥à â®à 

Q

. ®á«¥¤­¥¥ ®§­ ç ¥â, çâ® limsup_t !+0 h

Q

(

v

+

th

)

;h

i0 8

h

2

X:

(3.3) ®ª ¦¥¬, çâ® ¨§ (3.3) á«¥¤ã¥â ¯à¨­ ¤«¥¦­®áâì

v

¯à®áâà ­áâ¢ã

W

2 q(). ‡ ¬¥â¨¬, çâ® ¤«ï «î¡®£®

h

2

X

áãé¥áâ¢ã¥â limsup t!+0 h

Q

(

v

+

th

)

;h

i = ((

Av

+

th

)

;h

);(

p;h

) + R lim t!+0

g

(

x;v

(

x

) +

th

(

x

))

h

(

x

)

dx

. ®« £ ï (

x

) = maxfj

g

(

x;v

(

x

)+)j

;

j

g

(

x;v

(

x

);)j g

;

(3.4) ¯®«ã稬 ¨§ (3.3)

g

(

h

) = (

Av;h

);(k

p

kY+k kY)k

h

kY 8

h

2

X:

Œ­®¦¥á⢮

X

¢áî¤ã ¯«®â­® ¢ ¯à®áâà ­á⢥

Y

, ä㭪樮­ «

g

(

h

) «¨­¥©­ë© ­ 

X

,   ¢ ᨫ㠯®á«¥¤­¥£® ­¥à ¢¥­á⢠ ®£à ­¨ç¥­ ­ 

X



Y

. ®í⮬ã

g

¤®¯ã᪠¥â ¥¤¨­á⢥­­®¥ ¯à®¤®«¦¥­¨¥ ¤® «¨­¥©­®£® ®£à ­¨ç¥­­®£® ä㭪樮­ «  ­ 

Y

. ‘«¥¤®¢ â¥«ì­®, áãé¥áâ¢ã¥â

z

2

Y

 =

L

q() â ª®©, çâ®

g

(

h

) =h

z;h

i 8

h

2

X

. ® íâ® ®§­ ç ¥â, çâ®

v

| ®¡®¡é¥­­®¥ à¥è¥­¨¥ § ¤ ç¨

Lu

(

x

) =

z

(

x

),

x

2,

Bu

j ; = 0, ¨§ 祣® á«¥¤ã¥â ¯à¨­ ¤«¥¦­®áâì



¯à®áâà ­áâ¢ã

W

2 q() ¨ à ¢¥­á⢮ ­ã«î á«¥¤ 

Bv

(

x

) ­  ; ( à£ã¬¥­â æ¨ï â  ¦¥, çâ® ¨ ¯à¨ à áᬮâ७¨¨ § ¤ ç¨ (3.1){(3.2)). —â®¡ë ¯à¨©â¨ ª ¯à®â¨¢®à¥ç¨î, ¯®áâந¬

h

2

X

, ¤«ï ª®â®à®£® lim t!+0 h

Q

(

v

+

th

)

;h

i

<

0. ’ ª ª ª mes+(

v

) 6 = 0 ¨ ¤«ï ãà ¢­¥­¨ï (0.1) ¢ë¯®«­¥­®

Ay

, â® áãé¥áâ¢ãîâ

i

2

I

¨

" >

0 â ª¨¥, çâ®

­¥­ã«¥¢®© ¡ã¤¥â ¬¥à  å®âï ¡ë ®¤­®£® ¨§ ¬­®¦¥á⢠1 = f

x

2j



(

x

) =

'

i(

x

)

; L'

i(

x

) +

g

(

x;'

i(

x

););

p

(

x

)

> "

g

;

2 = f

x

2j



(

x

) =

'

i(

x

)

; L'

i(

x

) +

g

(

x;'

i(

x

););

p

(

x

)

<

;

"

g

:

ãáâì ¤«ï ®¯à¥¤¥«¥­­®á⨠mes1 =



6 = 0. ’ ª ª ª ä㭪樨

Lv

(

x

),

p

(

x

) ¨ (

x

), ®¯à¥¤¥«¥­-­ ï à ¢¥­á⢮¬ (3.4), á㬬¨àã¥¬ë ­  , â® ­ ©¤¥âáï

 >

0 â ª®¥, çâ® ¤«ï «î¡®£® ¨§¬¥à¨¬®£® ¬­®¦¥á⢠

B

, ¬¥à  ª®â®à®£® ¬¥­ìè¥



, ¢¥à­ë ­¥à ¢¥­á⢠ Z Bj

Lv

(

x

)j

dx < "=

8

;

Z B (

x

)

dx < "=

8

;

Z Bj

p

(

x

)j

dx < "=

8

:

(3.5) ‘ãé¥áâ¢ãîâ § ¬ª­ã⮥ ¬­®¦¥á⢮

F

 1, ¬¥à  ª®â®à®£® ¡®«ìè¥

=

2, ¨ ®âªàë⮥ ¬­®¦¥á⢮

G



F

, § ¬ëª ­¨¥ ª®â®à®£® ᮤ¥à¦¨âáï ¢ , â ª¨¥, çâ® mes(

G

n

F

)

< 

[14]. ãáâì

H

| ¡¥á-ª®­¥ç­® ¤¨ää¥à¥­æ¨à㥬 ï äã­ªæ¨ï ­  , à ¢­ ï ¥¤¨­¨æ¥ ­ 

F

, ­ã«î ¢­¥

G

¨ 0 

H

 1 ­ 

G

n

F

. ˆ§ ᮢ¯ ¤¥­¨ï



(

x

) ¨

'

i(

x

) ­ 

F

á«¥¤ã¥â, çâ®

Lu

(

x

) =

L'

i(

x

) ¯®ç⨠¢áî¤ã ­ 

F

([25], á.151). Žâá ¤«ï

h

(

x

) =;

H

(

x

) ¯®«ã稬 lim t!+0 h

Q

(

v

+

th

)

;h

i= (

A;h

);(

p;h

) + Z lim t!+0

g

(

x;v

(

x

) +

th

(

x

))

h

(

x

)

dx

= =; Z F(

L'

i(

x

) +

g

(

x;'

i(

x

););

p

(

x

))

dx

+ Z GnF

L

(

x

)

h

(

x

)

dx

+ +Z GnF lim t!+0

g

(

x;v

(

x

) +

th

(

x

))

h

(

x

)

dx

; Z GnF

p

(

x

)

h

(

x

)

dx:

’ ª ª ª

F

 1,   mes(

G

n

F

)

< 

¢«¥ç¥â (3.5), â® lim t!+0 h

Q

(

v

+

th

)

;h

i

<

;

"

mes

F

+ 3

"=

8

<

;

"=

2 + 3

"=

8 =;

"=

8

<

0

:

®«ã祭­®¥ ­¥à ¢¥­á⢮ ¯à®â¨¢®à¥ç¨â (3.3). …᫨ mes1= 0, â® mes2=



6 = 0. Š ª ¨ ¢ á«ãç ¥ mes1 6 = 0, ¢ë¡¨à ¥âáï

 >

0 â ª®¥, çâ® ¤«ï «î¡®£® ¨§¬¥à¨¬®£® ¬­®¦¥á⢠

B

 á mes

B < 

¢¥à­® (3.5), ¨ áâநâáï äã­ªæ¨ï

H

(

x

) á § ¬¥­®© 1 ­  2. „ «¥¥, ¯®« £ ï

h

(

x

) =

H

(

x

), ¯®«ã稬 lim t!+0 h

Q

(

v

+

th

)

;h

i

<

;

"=

8

<

0

;

çâ® ¯à®â¨¢®à¥ç¨â (3.3). ˆâ ª, ãáâ ­®¢«¥­®, çâ®

Ay

¤«ï ãà ¢­¥­¨ï (0.1) ¢«¥ç¥â ॣã«ïà­®áâì â®ç¥ª à §à뢠 ¤«ï ®¯¥à â®à 

Q

. ‡­ ç¨â, ¢ ᨫã ⥮६ë 1.1 «î¡®¥

u

0 2

X

, ¤«ï ª®â®à®£®

'

(

u

0) = inf_X

'

(

u

), ï¥âáï à¥è¥­¨¥¬ ãà ¢­¥­¨ï (1.1) ¨ â®çª®© à ¤¨ «ì­®© ­¥¯à¥à뢭®á⨠®¯¥à â®à 

P



TP

. Žâá á«¥¤ã¥â, çâ® â ª®¥

u

0 | ®¡®¡é¥­­®¥ à¥è¥­¨¥ § ¤ ç¨ (0.1){(0.2) ¨ lim t!0 Z

g

(

x;u

0(

x

) +

th

(

x

))

h

(

x

)

dx

= Z

g

(

x;u

0(

x

))

h

(

x

)

dx

8

h

2

X:

(3.6) Š ª ¨ à ­ìè¥, ¯®«ãç ¥¬, çâ®

u

0 | ᨫ쭮¥ à¥è¥­¨¥ § ¤ ç¨ (0.1){(0.2) ¨§

W

2 q(). ˆ§ (3.6) á«¥¤ã¥â, çâ®

u

0 | ¯®«ã¯à ¢¨«ì­®¥ à¥è¥­¨¥ § ¤ ç¨ (0.1){(0.2). „¥©á⢨⥫쭮, ¢ ¯à®â¨¢­®¬ á«ãç ¥ mes(

u

0) 6 = 0 ¨, §­ ç¨â, ®â«¨ç­  ®â ­ã«ï ¬¥à  å®âï ¡ë ®¤­®£® ¨§ ¬­®¦¥á⢠+(

u

0), ;(

u

0). „®¯ãá⨬, çâ® mes+(

u

0) 6 = 0. ’®£¤  ­ ©¤ãâáï ¯®«®¦¨â¥«ì­ë¥ ç¨á« 

"

¨



, ¤«ï ª®â®àëå ¬¥à  ¬­®¦¥á⢠ 1= f

x

2j

g

(

x;u

0(

x

)+) ;

g

(

x;u

0(

x

) ;)

> "

g à ¢­ 



. ‚롥६

 >

0 â ª, çâ® ¤«ï ¯à®¨§¢®«ì­®£® ¨§¬¥à¨¬®£® ¬­®¦¥á⢠

B

 ­¥à ¢¥­á⢮ mes

B < 

¢«¥ç¥â (3.5). „ «¥¥, ­ ©¤ãâáï § ¬ª­ã⮥ ¬­®¦¥á⢮

F

 +(

u

0) á mes

F > =

2 ¨ ®âªàë⮥ ¬­®¦¥á⢮

G



F

, § ¬ëª ­¨¥ ª®â®à®£® ᮤ¥à¦¨âáï ¢ , â ª¨¥, çâ® mes

G

n

F < 

[14]. ãáâì

h

(

x

) | ¡¥áª®­¥ç­®

¤¨ää¥à¥­æ¨à㥬 ï äã­ªæ¨ï ­  , à ¢­ ï ¥¤¨­¨æ¥ ­ 

F

, ­ã«î ¢­¥

G

¨ 0 

h

(

x

)  1, ¥á«¨

x

2

G

n

F

. ”ã­ªæ¨ï

h

2

W

1 2() ¨ lim t!+0 Z

g

(

x;u

0(

x

) +

th

(

x

))

h

(

x

)

dx

; lim t!;0 Z

g

(

x;u

0(

x

) +

th

(

x

))

h

(

x

)

dx

  Z G(

g

(

x;u

0(

x

)+) ;

g

(

x;u

0(

x

) ;))

dx

; Z GnF 2 (

x

)

dx



"=

2;2

"=

8 =

"=

4

>

0

;

çâ® ¯à®â¨¢®à¥ç¨â (3.6). ‘«ãç ©, ª®£¤  mes;(

u

0) 6 = 0, à áᬠâਢ ¥âáï  ­ «®£¨ç­®.   í⮬ § ¢¥àè ¥âáï ¤®ª § â¥«ìá⢮ ⥮६ë 1.3 ¢ á¨âã æ¨¨, ª®£¤  ¤«ï ãà ¢­¥­¨ï (0.1) ¢ë¯®«­¥­®

Ay

. ãáâì ⥯¥àì ¤«ï ãà ¢­¥­¨ï (0.1) ¢ë¯®«­¥­®

A

1

y

,

u

0 2

X

¨

'

(

u

0) = inf X

'

(

u

). Š ª 㦥 ãáâ ­®¢«¥­®,

u

0 2

W

2 q() 㤮¢«¥â¢®àï¥â £à ­¨ç­®¬ã ãá«®¢¨î (0.2) ¨ ¢ª«î祭¨î (1.11). „«ï ¯®ç⨠¢á¥å

X

2n(

u

0) ®â१®ª [

g

;(

x;u

0)

;g

+(

x;u

0)] = f

g

(

x;u

0) g, ¨, §­ ç¨â,

Lu

0(

x

)+

g

(

x;u

0(

x

)) =

p

(

x

). „®ª ¦¥¬, çâ® mes;(

u

0) = 0. „®¯ãá⨬ ¯à®â¨¢­®¥, ⮣¤  ®â«¨ç­  ®â ­ã«ï ¬¥à  ®¤­®£® ¨§ ¬­®¦¥á⢠; 1 = f

x

2 ;(

u

0) j

Lu

0(

x

) +

g

(

x;u

0(

x

)+) ;

p

(

x

)0g

;

+ 2 = f

x

2 ;(

u

0) j

Lu

0(

x

) +

g

(

x;u

0(

x

)+) ;

p

(

x

)

<

0g

:

ãáâì ¤«ï ®¯à¥¤¥«¥­­®á⨠mes; 1(

u

0) 6 = 0, ⮣¤  ­¥ à ¢­  ­ã«î ¬¥à  ¬­®¦¥á⢠ 1(0), £¤¥ 1(

"

) = f

x

2 ;(

u

0) j

Lu

0(

x

) +

g

(

x;u

0(

x

) ;);

p

(

x

)

> "

g

:

®í⮬ã áãé¥áâ¢ã¥â

" >

0, ¤«ï ª®â®à®£® ®â«¨ç­  ®â ­ã«ï ¬¥à  ¬­®¦¥á⢠ 1(

"

). ®«®¦¨¬ mes1(

"

) =



.  ©¤¥âáï

 >

0 â ª®¥, çâ® ¤«ï ¯à®¨§¢®«ì­®£® ¨§¬¥à¨¬®£® ¬­®¦¥á⢠

B

 ­¥à ¢¥­á⢮ mes

B < 

¢«¥ç¥â (3.5). ‘ãé¥áâ¢ã¥â § ¬ª­ã⮥ ¬­®¦¥á⢮

F

 1(

"

) á mes

F > =

2 ¨ ®âªàë⮥ ¬­®¦¥á⢮

G



F

, § ¬ëª ­¨¥ ª®â®à®£® ᮤ¥à¦¨âáï ¢ , ¯à¨ç¥¬ mes

G

n

F < 

[14]. ãáâì

H

(

x

) | ¡¥áª®­¥ç­® ¤¨ää¥à¥­æ¨à㥬 ï äã­ªæ¨ï ­  , à ¢­ ï ¥¤¨­¨æ¥ ­ 

F

, ­ã«î ¢­¥

G

¨ 0

H

1 ­ 

G

n

F

. „«ï

h

(

x

) =;

H

(

x

)2 

W

1 2() ¨¬¥¥¬ ¤«ï ¯à®¨§¢®«ì­®£®

t >

0 (

'

(

u

0+

th

) ;

'

(

u

0))

=t

= Z 1 0 h

Q

(

u

0+

th

)

;h

i

d:

¥à¥å®¤ï ª ¯à¥¤¥«ã ¯à¨

t

!0, ¯®«ã稬 lim t!+0

'

(

u

0+

th

) ;

'

(

u

0)

t

= (

Au

0

;h

) ;(

p;h

) + Z lim t!+0

g

(

x;u

0(

x

) +

th

(

x

))

h

(

x

)

dx

= =; Z F(

Lu

0(

x

) +

g

(

x;u

0(

x

) ;);

p

(

x

))

dx

+ +Z GnF (

Lu

0(

x

) ;

p

(

x

) + lim t!+0

g

(

x;u

0(

x

) +

th

(

x

)))

h

(

x

)

dx <

;

"=

2 + 3

"=

8 =;

"=

8

<

0

:

‘ ¤à㣮© áâ®à®­ë,

'

(

u

0) = inf X

'

(

u

), ¯®í⮬ã limt!+0 '(u 0 +th);'(u 0 ) t 0. ®«ã祭­®¥ ¯à®â¨¢®à¥ç¨¥ ¤®ª §ë¢ ¥â, çâ® mes; 1 = 0. €­ «®£¨ç­® ãáâ ­ ¢«¨¢ ¥âáï, çâ® mes+ 1 = 0. ‘«¥¤®¢ â¥«ì­®, mes ;(

u

0) = 0. Œ­®¦¥á⢮ +(

u

0) ¯à¥¤áâ ¢¨¬® ª ª ®¡ê¥¤¨­¥­¨¥ ¤¢ãå ­¥¯¥à¥á¥ª îé¨åáï ¬­®¦¥áâ¢: + 11 = f

x

2 +(

u

0) j

Lu

0(

x

) +

g

(

x;u

0(

x

)) ;

p

(

x

)6= 0g

;

+ 12 = f

x

2 +(

u

0) j

Lu

0(

x

) +

g

(

x;u

0(

x

)+) =

p

(

x

) g

:

…᫨ ¯à¥¤¯®«®¦¨âì, çâ® mes+ 11 6 = 0, â® ¨§

A

1

y

¤«ï ãà ¢­¥­¨ï (0.1) á«¥¤ã¥â áãé¥á⢮¢ ­¨¥

i

2

I

, ¤«ï ª®â®à®£® ¬¥à  ¬­®¦¥á⢠ i 11= f

x

2 + 11(

u

0) j

u

0(

x

) =

'

i(

x

)

;

(

L'

i(

x

)+

g

(

x;'

i(

x

)+) ;

p

(

x

))(

L'

i(

x

)+

g

(

x;'

i(

x

););

p

(

x

))

>

0g