các bất đẳng thức tích phân thuộc loại Ostrowski và các áp dụng của nó 2_2

f!lJal drb(? flute fk./' jtluiu

h,i

Trang 5

(j)~t~()rtOki

CIIUdNG I

CAC DANG THUC TicH PHAN
Mt,lcdkh cua chlidng ngy la tr'inh bay mQt s6 cac d~ng thuc tkh
phan bi~u di~n theo gia trj ham va cac dC;lOam cua no tren cac khming
h
tlidng ung. C6ng ct,lchu ye'"ula vi~c sa dt,lng chung minh qui nl,lpva mQt .
s6 c6ng thuc trong phep tinh vi tkh phan.
Tnioc he'"t, ke'"tqua sau day.
})jnh IX 1.1.
Cho

Ik

:a

= Xo < XI < ",Xk-I

< Xk

= b la mQt phep pluln ho{Zeh eua do{Zn
~

[a,b], aj (i=O,...,k+l)

va

ak+1

La "k + 2" .diem saG eho ao =a,a; E[X;_"X;] (i=l,...,k)

= b. Ntu /: [a,b] ~ IR c() d{Zo ham dtn dip 11 va /(11-1)lien t1;le
-I

tuy~ t di;'i trerl [a,b]. Khi de) ta 'eel dang tlu/e:
b

(1.1)

Jf(
a

t dt+~(-l)f~f
)
~

.;=1

(X

-a

.,~~/"'I

}.

fU-]) x1+] - (x-a
11+1.

.I

)
1+1

;=0

b

= (-1)"

JK,;,k(t)/(II)(t)dt,

a

trong do nluln Peano dU:(/e
eho b(Ji:
(t-alr,

11!

t E [a, x,),

(t-a2r, tEfxl,X2)'

11!

( 1.2)

KII,k(t) =
(/-ak_,)"
11!
-,
(f-akr
11!

, t E [Xk-2,Xk-I)'
.
IE[Xk-),b],

(

)

J U-I)
) f
(X )}
I


vcJi n,k E IN va f(°>Cx) lex).
=

Chung minh.
Ta chung minh b~ng qui n(;lptoaD hQC.

Vc3i 11p-1, chung to can cht1ngminhdang tl1l1c
h

f f(l)dt

(1.3)

.~.

k-I

L [(X;+I - a;+1 )f(X;+I)

-

a

- (x; - a;+1 )f(xj)]

;=0

h

=trong

J

a

KI,k (t)/I

(t)dl,

do
(I - al),

IE [O,XI)'

(t-a2),

IE[X"X2)'

(I-ak-I)'

IE[Xk-2,Xk-')'

(I -ak).

{E Ixk-I,bl.

K1,k(I) =

D~ chung minh (1.3), ta dung tich phan tUng phfin nhtl'sau
h

k-P" ,

f K1,k (/)( (/)dt
a

== L

f

K1,k

(t).t (I)dl

;=0 x;
k-Ix'+1

=

L J (t -a;+I)f

(t)dl

;=0 x;

= *i (t 1=0
[
~ ~[(x",

al+1

)/(t)

Ix;'1

x,

- xJ :l.
x,

]

- a",)[ (x"' ) - (x, - a", )[(x, ) - '[f(t)dt ]

k-l

=

k-I x,+,

I [(X;+I
-a,+I)/(x;+I)-(x;

~

-a;+I)/(x;)]-

I Jf(t)dl

~~

k-I

h

= L [(X'+I a,+1)/(X;+I
~

I>

Do do

I>

) - (XI - a,+1)f(x;)]

a

Jf(t)dl

a

k .-1

ff(l)dl + fK1,kI)f (t)dl = 2](X;+1
(
-

a

-

;=0

a;+1)f(x;+I)

- (X; - a;+J )f(xJ]

V~y d~ng thuc (1.3) dtl
(j).

,JU,!n

'f,- olZ

- ,if "fIJ

{/an ,H/(!(' ');'/ .,(Jan .7If)('

' f'::- 'fIJ"., (1),.J ?U,,/'!JI, .7(,(H( ::.l/((1'fI

I


Trang 7

(11(11dd!1,? 11t(f'c Iklt .la'trill, UJq;;CMt()l~ti

Giel sli' r~ng (1.1) du ng vdi "n" va ta c~n chung minh r~ng (1.1) dung vdi
"n+ 1", tuc la, ta chung minh d£ng thuc sau day dung:

"

'

k-I

//+.1

J/( t )d t + ~

a

(1.4)

,

'}

",,(-I)

f(

~ ~~
}=I
J,

;=0

X;+I -ai+I,

)

1

rU-I) ( X;+I ) - ( X; -a;+1

)

1

f U-')

( X; )}

"
= (-1)"+1 JKI1+I,k(t)f(n+I)(t)dt,
a

trong do nhan Peano Kn+l,k du'<;1c ho bdi:
(t)
c
(t -

al )"+1

(n+l)!

'

tE[a,xl)'

(t - a2 )"+1
(11+1)1 '

tE[XpX2)'

K n+l,k (t) =

(t -a k-l )

"+1

(n+l)!
(t -a

k)

' tE[Xk-2,Xk-I)'

/1+1

(n+1)!'

tElXk-j,bj.

Xet tich phan
It

k-I

JKI1+I,k
(t)f"+1\t)dt

=

a

X,+,

I

J KI1+I,k(t)f(I1+1) (t)dt

;=0

=

Xi

I

xJ (t ~a;+t)"+1 f(n+')(t)dt

;=0

x,

(n+l)!

sau do dung tich phan tUng ph~n ta du'<;1c:
It

J K//+I,k(t)P/HI)

a

(t)dt
k-I

I

= ;=()

I

[(

)

//+1

-a;+I.

f(//)(t)

(11+1)!

k-l

)

=I ;=0 X;+t-ai+\
(11+1).

[(
-

Xi>l

J

IX'" -

x,

//+1

Xi

(t

)
-a;+1
11!

)

(
f(I1)(Xi+I)-

IxJ (t -a;+I)"
;=0 Xi
n,

x;-a;+I,
(n+l),

f(I1)(t)dt

//

f(I1) (t)dt

"+1

f(I1)(x;)

]

]


k-t

'1+1

=L[( Xi+l-ai+"
'=0
(n+1).

)

(
j(I1)(XJ+I)-

)
Xi-ai+l,

j(I1)(X,)

(n+1).

,

,
I
i
i

"+1

"

]

- K (I) (I1) ( t )dt.
J I1.k '.

i

(1

Ta vie't l~i d~ng thuc ri§y
I

"
(1.5)

"

,,+1

'k..1

fK

(t )j

(I1)

l1ok

)
(t )dt =',L.. Xi+l-tXi+1
'
'1=0
[( ( n+. 1)
,

0

'

j

(I1)

(x

,,->1

(
)
) - X, -ai+1 1
1+1
(n+ 1).

j

(") X
( ')

I"

]

- JKI1+lok(t)/,,+1)(t)dt.
,0

Theo gia thie't qui n~p, ta co:

"

( 1)1 k-I

2::~2::{./=I.J.
i=O
11

Jj(t)dt+
(1

jU-I) (xJ}

"
= (-1)" JKl1ok(t)f(I1)(t)dt.
0

hay

"

( 1.6)

"

fK".k(t)j(I1)(t)dt = (-1)" Jj(t)dt

0

a

"

"

./ k-I
(- 1)
./
f(
1
+ (- ) .L.. ~ I L.. ~ X'+I -a'+1 ) j
I ./,"
.1
'IOn
/1"

U-I)

1
(Xi+1 - (Xi -ai+1 ) j
)

U-1)

(X, )}

Tir (1.5) va (1.6), ta co:
(-1)" a
fj(t)dt

+ (-lrI

1=1

-(Xi'-ai+I)'/

I

(-?/
.J.

i=O

(
jU-I\Xi+')

fU-')(X;)}

k-I

=2:: Xi+l-a'+I,
,=0[(
(n+l).

)

/+1

)

(
f(I1)(Xi+I)-

/+1

x,-ai+l,

f(/1)(xJ

(n+l).

"

- K (t )f
J 11+I.k

]

(I/+') t

( ) dt ,

a

hay
k-I
(
"" ~~1) f(

"

11

r

S J ( t )d {+ L..L..
a
,=0 ./=1 J.

(fl.

.,'-

If"qll

NTlI ./Tlqr

'j7/

- Ijl'

1

X'+I -ai+1

)

f

./
0

'U-')

1

( X'+I ) - ( Xi -ai+1 ) f

U-')

( Xi )}

'{/}

'),r .1'(JfT.1I.-nfll'.

,A:9'-"J/%n ,m«,

9/!r'hlfl


J.2
,7,Ja.1 a-aJ'fl
(j7J _

//,
",ute

/, /
/
"~JI. Iil/tan-

Trang 9

/
41 /
/
waf . W,jr,lm!ijh't .

k-I

'1+1

'

)

=(-1)"2: [( xi+l-ai+\
i=O
(n+l).

,

)

(
f(II)(Xi+I)-

'1+1

x;-a;+I,

l")(X,)

(n+l).

]

h

- (-1)/1 JK,1f1,k(l)f(II+I)(l)dt
a

hay
h

fr(t
.
a

;

""
k-I

)d

II

). f
( 1
(

t+ ~~~1Xi+l-ai+1
i=O1=1 J.

J
1
)
) f (;-')( Xi+1 - ( xi-ai+1 ) f

k-I

+ (_1)"+!

)

L
xi+,-a/+1
i=O
[( (n+l)!

'1+1

(

j(II)(Xi+1)-

(;-I)

)

(

Xi)}

'1+1

Xi -ai+1
(n+l)!-

j(II)(Xi)

]

h

= (-1)"+1

J
a

(I )j(I1+I)

KII+l.k

(t)dl

hay.

"" 1) f
Jr( )d + ~~ ~t (

h

(1. 7)

k-I

II

(-

a

.

1

1

;

J

.

Xi+1

,=01=1'/'

'

+(-l)"!'~
[(
(n+ 1). £..i
i=O

-

x -a
Ii"

ai+l.

1...1

)

)

"+'

f

f (;-I)
(/I)

( Xi+1 ) - ( Xi -

-

( x 1+1) ( x-a

1

ai+1

1+1

)

)

1

"+I

f

f

(;-I)

(II)

( Xi )}

( X 1)]

h

= ( _1 )

11+1

J

K II! l,k. (I ) f

(II+I)

(I ) dl..

a

sO' hc:lllgthu hai va thu ba cua v~ tnii vi~t gom l~i thanh mOt sO'he.mg,do
do ta thu duQc:
h

(1.8)

Jj()d
t

a

""

k-lll+1

1
(- 1) f

t+ ~~~tXi+l-ai+l.
i=O}=1 ./.

(

1

) f

(;-I)

1

(Xi+1 - (xi-ai+1 ) f
)

(;-I)

(Xi )}

h

= (-1)"+1 f~:11+1,k
(1)/(11+1)(t)dl,
a

nghla la, d~ng thuc (1.1) la dung va Dinh ly 1.1 duQc chung minh.H~ qua sau day cho mOt d~ng thuc tich phan khac vdi (1.1) se heru
ich trong cac ph§n sau.

ryMT
dr/,,//- 1/,,((' Iff-/' A/'rln f(J(d (il.}t;(J(t)k;

Htimi

Tran~lQ

1.1.
Vdi ClJnggid thief cua djnh (v 1.1, ta co :

~ (-;n~ tx, - a,)J
b
= (_1)11 J (l)f(l1) (I)di.
Kn,k
a
+

ff({)d{

(1.9)

- (x, -a",)J

}f(j-I) (X,)]

Chung minh.
Tli (1.1) ta xet s6 hC;lngthil hai va vie't no thanh tc5ng cua hai s6 hC;lng
8

.

1

+

8

n

2

J k-I

(- I) "I
(
- ~ -:--, ~ ~ Xi+1 J. i=O
/=1

-"

/

ai+1

) f

u-IJ

<

(Xi+1 ) - (Xi -

J

ai+1

) f

U-')

(Xi )}

k-l

(1.10)

= L{-(Xi+' -a'+I)f(Xi+I)+(Xi -ai+l)f(Xi)}
1=0

J k-I
( I) ,,

II

,,-

+ ~ -:--, ~ {(Xi+) /=2

}.

i=O

/

) f

ai+1

U-J)

<

( Xi+l ) - ( Xi<-

ai+1 )

J

f

U-IJ

(Xi )}

k-I

=:

L {1=0 <

(X'+I

""
k-I

/I

(-

- £Xl"!)f(Xi-l')
I) '
.

.

f

(
+L..L..~~X1+1-ai'l.
i=O
J=2 J.

Bay gio

+ (x, - £x/¥
/

) f

U-')

( xi+l-x,-ai+1
) (

81 du'..

81 = (a-a,)f(a)+

k-I

L(xi
i=1

-a'+I)f(xi)

k-2

+ 2)~(Xi+l
i=O

-ai+,)f(x,+,)}-(b-ak)f(b)
k-I

=(a-a)f(a)+

L(x;
;=1

-ai+,)f(xi)

k-I

+ L{-(xi
;=1

.

=(a-a,)f(a).

-aJf(xi)}-(b-ak)f(b)
k-I

L(ai+1 -aJf(xi)-(b-ak)f(b).
1=1

Cling v~y voi 82 du'
J

) f

U-')

(X;.
)}


,,=

k-I

<;
L

2

/I

L..JL..J
1=0/=2

k-I

II

'
( 1 f
)
--=--,t ( Xi+1
J.

;

.

) ..' [

-a,+1

(/-I)

(X;+I ) - ( X,

;

). f

-a;+1

U~')

( X; )}

/I (-1)/

= i=0)=2

--=--,{(Xi+1 -a;+YfU-I)(Xi+I)}

J.

II
= I (-~r

.

(-~r {ex;-al+YfU-l)(xJ}

-

i=O
)=2 J.

)=2

J.

II (-~r

{(Xk-ak)/ fU-I)(Xk)}+

i=O)=2

J.

~~ (-Jr {(Xi-a'~I)' IU-I)(x,)}

- ~ (-;({(XU-aYIU-I)(xU)}-

= I(-~r {(Xk-ak») fU-I)(Xk)}+II(-~r
)=2 J.
,=1)=2
-

I (-~r
J.

)=2

I

= 1=2(-~r
j.
+

{(Xu -al»)

= t(-?/
)=2 J.

fU-I) (Xo)}-

{(Xk-ak») fU-')(Xk)}-

~[~ (-X

II
I
/=2

{(Xk -ak»)fU-I)(Xk)-(XO

I [I (-~r
)=2

J.

(-~r {(Xi-ai+I») fU-') (X;)}

(-~r (exo-a,») fU-l)(xo)}

J.

-al»)

.

{(X;-aY

fU-')(XO)}

-(x, -aH,Y V"-l) (X,)]

I

'=1

(eX,-a,») fU-I)(Xi)}

;=1)=2 J.

= )=2 (-~r {(b-akY fU-I)(b)-(a-a,y
J.
+

J.

{(x,~a,)i ~(x, ~a",)j IF"IX,)]

~[~ (~;t {
+

-a;+,») jU-I)(X'+I)}

(cXi+1

-(X;

-a,+I»)

fU-')(a)}

}JU-I) (X;)

.
]

Tli (1.10), ta co
k-I

8, +S2

= (a-a,).!(a).
+

I (-~r
J.
)=2

I(a,~,-a,).!(x;)-(h-ak).!(h)
;=1

..-------

r

'1:)~.kH.T(JN~IEN
{(b -ak») fU-')(b)

- (a -al») j<)-I)(a)}

THLT
\lIEN

-

OOO~86

J


[1M1 rlrt/Ift (/",,(' (fr/'Ji/uin

+

~

%[~

(ii'

kx,-a,J' - (x,-a",dfU-"(x,)

I

-{(a, -a)f(a)

+ ~(a",

I

+ 1=2 (-?l [-(a-a,)}
J.
+

Chli

Y

r~ng

I

,=1

Trang 12

!.Jai @.)b(J,~t,,:

(eXi~a,)}

-a,)f(x,)

]

+ (b -a,)f(b)}

fCj-I) (a)

-(Xi -ai+I)}}JU-I)(xJ+(b-akf

fU-')(b)]

a, ao = a, Xk = b Va ak+1 = b ta co th€ vie"t

Xo =

k-I

8, +82 =- { (al-a)/(a)+

t;(a,+,-a,)/(xJ+(b-ak)f(b)

}

+ I(-?i J. [-(a-al)}/C!-I)(a)
}=2
+

I

i=1

{(x, -aJi

-(x, -a1+,)}}fU-')(x,)+(b-ak)}fU-')(h)}

k

=-I(a'+1

-a,Jf(x,J

i=O

I

+ }=2(-~r [-(a-at)}
J.

fU-')(a)

k-I
+

I

;=1

{(x; -ai)}

-(Xi -ai+I)}}/U-I)(x;)+(b-ak)}

fU-1) (b)]

k

=-I(ai+1
;=0

-a;)f(xi)

+ I(-~r [-(xo-a,YfCH)(xo)
i=2 J.
k-I

+

I (ex,-aJ/-(x,
,=1
k

= - I(aj+1
,=0

'

-aj)f(x;)
k

11 ( 1)i

+

I~

.1=2 J.

-aj+,y}rU-')(x;)+(Xk -akYfi-I)(xd]

I

[ ,=0{(Xj-aJl
,
,

-(x; -a;+,f}fu-')(xJ

]


Trang 13

[1],;-1UfJ~rIluf-clicit ./tltalt kat (lM~()(':-)/{i
t

~t

[t kx, -a,)'

(-;:J

- (x, -a", Ji}J"O"(X,)}

Thay 8, + 82 vao so' h~lOgthU'hai cua (1.1) ta tho duQc d~ng thU'c(1.9).
Bay gio ta gia Slt rang cac di~m chia Xi cua phan ho~ch

la ceSdinh,

lk

ta

tho duQc h<$qua gall.

He !loa 1.2.
Cho lk :a=xo Ne'u f: [a,b] --+IR gi(fng nhll tron!? dinh Ly 1.1. Khi d6 ta c6 &lng thac:
k

hill

(1.11)

ff(t)dt+

+(-l)Jh/-I}rU-I)(x;) ]

~2Jj! [ ~{-h/

h

= (-1)"

fKII,k (t)f(lI)

(t)dt,

a

trong d6 hi = X'+I- x" h_1= 0 Va hk = O.

Chung minh.
Ch<;>ncac di~m a; U=O,...,k+l)
,

ao=a,al=-,a;=

ak =

Xk-I

a+xl

nht!sau:

Xi-I +X,

2

+ Xk
2

2

.

, (z=l,...,k),

'

va ak+' = b.

D~ng thU'c(1.9) vi€t l~i
h

(1.12)

h

ff(t)dt+81

+82 =(-1)"

a

vdi

8, + 82 =

f

/=1

(-

?J

}!

i:

[ i=O

kXi - ai)/

fK",k(t)f(II)(t)dt,

a

- (Xi - ai+I)/

}fU-'J

(Xi )

.

]

Ta chia so' h~ng thil hai 81+ 82 cua (1.9) thanh 3 so' h~ng nhu gall:


~Jnlitj{(

'uoAl1I!iJJ&ui.f1JlI}(L
(~

_.Imugjj

- "",,-," "'-~

k

(])./
-

11

.

,

SI+82=~-j!~

[

(-I)

,

}

k

xi-a;)./-,(Xi-ai+t)./fJ X;)
( ]

k

'

= -2: {(Xi -a;)-(Xi

-ai+I)}[(Xi)

;=0

+

f

(-?}

./=2

+

kb-adJ

fU-')(b)-(a-aj)}

fU-I)(a)}

.1.

~[t I-~:j ¥X;- aY

sO' h~ng thil nha't 8, +82 cua (1.9)

- (X;- a;+I)'

Ir(JI)(X;)

0 ~ i~ k, .i

vdi

J

= 1:

k
(i)

:L{(x,-a,)-(x,-a,+,)}f(xJ
'=0
k-I

=-(a-a,)f(a)+(b-ak)f(h)+

I{(i,-a,)-(x,-a/+,)}f(x,)
1=1

-

1

f

k-I

- 2lhof(a)

S6h~ngthuhai
(ii)

i=O,i=k,j~2:

I(-~r.1. (cb-ak)} fU-')(b)-(a-a,)J
}=2
L. 2 J
J=2.1.
'I

S6h~ng

{

hlrU-I)
k I.

}.

+ h,-,)f(x,) + hk-J(b)

81+52 cua(1.9)vdi

= ~(_])I

(iii)

+ ~(hi

fU-I)(a)}
)}.

( b) - ( -I )JhllU-I)(a
O.

thil ba 8, +52 cua (1.9) vdi ]~i~k-l,

2~.i~n:

~ [~( -;r {(x, aY - (x,- a,.,)'}Iii "(x,)]
=

I i:(~
'

1)ij h~1

[ 1~2.1 !2

I~'

-(-I)J h/lrCH) (XI)

,

'

]

f)~t ba sO'h~ng (i ), (ii ) va (iii ) vaa 81+ 52 ta thu du'Qc:

(Ll3) S, + S,

= ~ (-;n~ «X;-a,J' -(X, -a",)' }/U"'(X,J]
k

= - ,=0{(x, -aJ - (x, -a;+,)}f(x;)
I


iYJril rI,j:'~11/ut(' lid" j'/uiu

Trang 15

(wi f~JIt()r(;jt;

i

+ J=2 (-?J {(b-adlf(l-I)(b)-(a-al)J
./.
k-I

n

J
)
(- 1

[

+~~fl(Xi-ai)

= -2

~

+

L.

J
-(xi-ai+i)

k-I

{ ho/(a) +
hJ{

(-l)J

J=2./.

+

J

{

1

'1

2l

1=1 l=2

=~

J.

~ (-l)l

fU-I)(b)-(-l)JhJ

{h~
/1

~ h~

= ~ (-1)J

(j-I)

(Xi)]

~ (hi + hi-I).I (x;) + hk-d(b)

{h;~~1

L. J=I J.2 l
L. 'I
1=0 [

}
/

,

k 1

I [ i (~;:

fU-') (a)}

{

L."2lL.11 1=0

J=I J.

0,

- (-l)Jh/
-(-l)J

-(-l)JhJ

fU-') (a)

}

lr(j-1) (x/) ]

hJ }jU-1)(X)
1

I.

}

/

]

}rU-1)(x).I

Cu6i cling ta thu dt(Qc (1.11) b~ng cach thay 81+82 vao (1.12)..

Tn(on g ho p ta lay cac di~m ehia x- eua Ik cach d~u , ta tlm duoe he q Ua
.
1
..
salt:

He qua 1.3. Cho
(1.14)

la mQt

Ik

: Xi

= a +:{b~ a). i = 0,...,k

phfin hOi;lChd~u cua doi;ln [a,b] Va f: [a,b] ~ IR sao

cho

In-I)

li@nt\,lctl1yi$td6i tr~n [a,b]. Khi d6 ta c6 dAng thd'c:

(1.15)

J/(t)dt +

t(

.I~I

0

.I

b -a
2k

)

x;,[ - Iu "(0) +
h
= (-1)"

~

I<-I)'

-I

}I"-"(x,) + (-1)1IU-'>(b)]

J KII,k (t).T(II) (t)dt.
11

Chu

y r~ng

.

sO'hi;lng thli hai cua (1.15) chI chlia cac di;lo ham cffp Ie ti;li

tfft ca cac diem trong
Chung minh.

Xi'

i = 1,...,k-1.

Sa d\,lng (1.14), ta chu y r~ng


:YM1
d(j,~1

b-a

ho =x! -x() =-,

va

TranR-16

(I"f'(' (ir-/' .;'/,(j,J/ !oa; (iM~()fr:Jt;

b-a

hi =xi+! -x/:::::-

k

k

b-a
hi!

=Xi -XI-I

""k,(i=t,...,k-l)

va the' vao (1.11), ta co:

"

( 1.16)

II

fI(t)dt +

b

I ( ~2k
.1=1

(1

.x

.1
.

)

;![ -f'HI(a)+

= (-1)"

"

~ k-l)'

-1)I'HI(x,)+(-I)J fU-"(b)]

fKII,k(t)f(II)(t)dt.

a

Tinh tacin don gian, tU day ta suy tu d~ng thuc (1.15).8
Cong thuc gi6ng Taylor sail day voi phfin du tich phan cling dung.
He gmi 1.4.
Cho X : [a,y] ~ 1R OJ d~w helm din clip n sao cho g(lI) lien t~lCtuy~t
at)'l tren ju,y]. Khl d6, wYi mql XiE [a,y) fa c6 dilng tlllJC:
(1.17)
g(y)

~

g(a) -

~

[t

(-;r

(lx", - a",)' g"'(x",) - (x, - a,.,)' g(j)(X,)}]

y
+(~1)" fKII,k(y,t)g(II+I)(t)dt
(1

hay

(1.18)

II (-1).1

g(y) = g(a) -

~ fl

k

[ t=o {(Xi- aJ1 - (Xi -ai+l)l }g(J)(xJ ]

y
+(-1)"

JKII,k(y,t)g(II+)(f)dt.
a

Chung minh cua (1.17) va (1.18) dlivc suy tn!c tie'p tu (1.1) va (1.9) Hin

luvt b~ng each clwn f = gl, b:::::
y..