TADEUSZ MARIAN JĘDRYKA Bydgoszcz
AN ESTIMATION OF THE SOLUTION OF VOLTERRA'S INTEGRAL EQUATION FOR VECTOR-VALUED FUNCTIONS WITH VALUES IN
SOME FUNCTION SPACES
I n [ 2 ] we d e a lt w ith a V o lt e r ra in t e g r a l equ ation
( 1 u (x ) ||∫aX T (x,t ) u ( t ) d t + b (x ) ,
where x , t , a
ϵ
Rn ; b and u are v e c to r -v a lu e d fu n c tio n s o f the va r i a b l e t (a ≤ t ≤ х ) w ith va lu es in a Banach space Y, and T ( x , t ) is a l in e a r bounded o p e ra to r o f Y in t o i t s e l f f o r a ≤ t ≤ x , s t r o n g ly measurable in both v a r ia b le s .The order r e la t io n c ≤ d f o r с = ( с , . . . c n ,) ϵ Rn ,d= ( d 1, . . . , dn) € Rn means here th a t ci≤di f o r i = 1 , 2 , . . . , n . Denoting by νγ the space o f a l l such o p era to rs and supposing th a t ||b (x)||Y ≤ B (x) where B (x) i s measurable and bounded f o r x≥ a , th ere was proved the f o llo w in g theorem :
T h e o r e m . Let us suppose th a t A ( x , t ) i s a r e a l- v a lu e d fu n c tio n , d e fin e d f o r a ≤ t ≤ x , non decreasin g w ith re s p e c t to x f o r eve ry t and such th a t A ( t , t ) i s measurable and bounded f o r t ≥ a ;
x
∫ A ( t , t ) d t < 1 f o r x≥ a . M oreover, l e t us suppose th a t a( 2 ) || T (x ,t )| |y≤
A ( x , t ) f o r a ≤ t ≤ x ,where
i s independent o f x and t . Then the in t e g r a l equ ation ( l ) has a unique s o lu tio n s in th e space o f a l l Y -valu ed bounded72
-and s t r o n g ly measurable fu nct io n s in x ≥ a. Moreover we have an e s tim a tio n
||
u(x)
|| y ≤ β(x)ex
P (∫
a x
A ( t , t ) d t)f o r x≤ a , where β ( x ) = sup B (x ) , and where ||b(x)|| y≤ B (x ) .a ≤ t ≤ x
1.The aim o f t h is s e c tio n i s t o v e r i f y the in e q u a lit y (
2
) in th e case when the o p e ra to r T (x , t ), w h ic h w i l l be b r i ef l y deno ted by T, i s g iv e n by th e e q u a lity v = Tu, where Vg. =So
aT
gτ,
du τ
and Y i s the space CVp o f co n tin u ous fu n c tio n s u o f bounded p - v a r ia t io n Vp (u ) in < a ,b > , p 1. L et us r e c a l l th a t th e p - v a r - ia t i o n o f u in < a ,b > i s d e fin e d by
V p(u ) = s u p ^ l u ^ - u ^ i - ! ^ ,
where
: a =τ
0<τ1
< . . . <τ n
= b i s an a r b it r a r y d iv is io n of thei n t e r v a l < a , b > (see L.G.Young, [ 5]) , Vp w i l l mean the space of a l l fu n c tio n s u f o r which Vp(u )< + ∞ . Then ||u||p = ( V p (u )1/P i s a seminorm in both, in CVp and Vp and||u||p = 0 i f and o n ly i f u i s
, ,
con stan t i n < a , b> . L.C.Young proved, [
5
] , th a t i f p, q > ^ » ~ and uϵ
CV , wϵ
V , whan th e R ie m a n n -S tie ltje s i n t e g r a l Sa wduP q b e x is t s and
S W 'dUj~ MHIqHp^p + q) ”here
"fe" *
We s h a ll prove now the fo llo w in g theorem :
Theorem 1. Let p,q > 1 , >1 and l e t a r e a l- v a lu e d fu n c tio n Tg,
τ
. d e fin e d f o r ,τϵ
a ,b > s a t i s f y the f o llo w in g c o n d itio n s :<(a ) T,
ϵ
V and T= 0 f o r e v e ry (ϵ
< a ,b > , Ta,. = 0,(b) the v e c to r-v a lu e d fu n c tio n f : < a , b > → Vq d e fin e d by f ( 6^) = Tg, , is continuons, in < a ,b > ,
v - 1 V du<c
i s a lin e a r continuous operator in CV and i t s norm P
I M I ^
with
< * = К
$(? ' D
-Proof. From the condition (b) i t fo llo w s that V = Vg is a con tinuous function in < a ,b > , because f o r a rb itra ry ^ , 6 ^ Є <a»b> we have
К ■
' J
S
^
V
V
^
Now, we have1M |p
'lr
M
( v -
ч.,.'Ыр
}}:<
.. *iv i,4V - Ч Л 'К ? (i * 0 " ■
= KAÇ ( p + q ) H p *
This completes the proof. —
2 .Now,keeping the operator T in the f orm v»Tu with
le t us consider the case if Y =V1 where V1 is the space of fun ctions of bounded v a ria tio n ||u| = V 1 ( u ) i n < a ,b > . The follow ing theorem holds:
- Theorem 2. Let T . be a re a l-v a lu e d function defined fo r £-ф ,Ь > and sa tisfy in g the follow ing conditions:
- 74
(b ) Т„» s a t i s f i e s the L ip s c h itz c o n d itio n w ith re s p e c t t o f or eve ry <a, b)> :
w ith the constant A independent o f f .
Then the operator T d efin ed as in Theorem 1 is a continuous lin e a r operator in V1 and i t s norm |jT|)^WA w ith c C = b - a .
P r o o f. We have
H i , - s^ p i I V "
\ ^ ) Ы - к(-ь
■ a) l (u " i *3. F in a lly , we are go in g t o estim ate the nora||Tj|^in case of the space Y = H o f fu n ctio n s s a t is fy in g the Höld e r c o n d itio n with an exponentγ , 0 < γ ≤ 1, i n the in t e r v a l < a , b>, the norm in H is d efin ed by
...
K ' - v 4
, u , H -
There holds
Theorem 3. Le t T „ be a re a l-v a lu e d fu n c tio n d efin ed f o r <a, b > and such that
(a) T i s an in te r g r a b le fu n ctio n o f i n < a , b> f o r e v e r y (a,b) ( b) T s a t i s f i e s the Hö ld e r co n d itio n
1Ь ' У
** Н
^ Г ' 1Г
-%fo r every <a , b>, w ith A independent o f t .
Then the operator T d e fin e d a s v = Tu w ith vTg-.= $ u ^dt i s a continuous lin e a r op era tor in H w ith norm
where
c * = ( b - a ) max(l, .
!|т|!н VS*
".
^ b ь SUP С T " 1/---1U ! dt ^ A C j u I d't .r><r*
ButKl ^ k a)i
[1
-(*- af l + iNhC* - аУ
fo r ever y £ £ 4 i,b > . Hence /.— лГ» /. v f jN H
REFERENCESC‘ - * ) “ » ( ; , -
j n
Ą M
h
•
[ l ] Z.B u tlew sk i; Sur la lim it a t io n d es so lu tio n s d'un système d 'e q u a t ions in té g r a le s de V o lte rra , Ann. Poion. Math., 6 1959, ■ p . 253-257
[
2]
T.M .Jędryka; Oszacowanie rozw iąza n ia równania са ł кowego V o l-t e r r y dla fu n k c ji wekl-torowych, Rocz. P o l. Tow. Mal-tem. Prace Matemat., I X 1965, p . 267-271[
3]
J . Musielak, W. Or l i c z ; On g e n e ra liz e d v a r ia t io n I , Studia Math. 18. 1959 p . 11-41|VJ T .S a to ; Sur la lim it a t io n des s o lu tio n s d'un Systeme d'equ a-tio n s in te g r a le s de V o lte r r a . Tohoku Math. J . 4.1952.p .272-274 L .C. Young; An in e q u a lity o f the Höld e r type, connected with S t i e l t j e s in te g r a t io n . Acta Math. 67.1935.251-281
[ 6 ] L .C .Young; General in e q u a lit ie s f o r S t i e l t j e s in te g r a ls and the convergence o f P o u rie r s e r ie s . Math.Annalen 115.1938. p . 581-612
V
7 6
-AU ESTIMATION OF THE SOLUTION 0F VOTERRA'S INTEGRAL EQUATION FOR VECTOR-VALUED FUNCTIONS WITH VALUES IN SOME FUNCTION SPACES
SUMMARY
There are given p ro o fs o f three theorems in t h is paper, in which i s d efin ed the constante o f the estim a tio n of the s o lu tio n o f V o lt e r r a 's in t e g r a l equation when the valu es o f the k ern el are in the CVp and V1 spaces and a ls o in case when the k e rn e l s a t is fy the Höld e r co n d itio n rega rd to param eter.
