Nguyen Thj NgSn T j p chl KHOA HOC & CONG NGH$ 99(11)' 169- 175 S O L V A B I L I T Y O F S Y S T E M S O F D U A L I N T E G R A L E Q U A T I O N S
I N V O L V I N G F O U R I E R T R A N S F O R M S N g u y e n T h i N g a n " , C o l l e g e o f E d u c a t i o n - T N U
S u m m a r y . T h e aim of t h e present p a p e r is to consider the solvability and solutions of systems of dual integral equations involving Fourier transforms. T h e uniqueness and existence theorems are proved in appropriate Sobolev spaces. We shall consider the cases: the case A ( 0 G S+{K) and the case A({) e EJCK).
Key words: Fourier transform, systems of dual integral equations, mixed boundary value problems.
1 Introduction
Dual integral equations arise in many areas of mathematical physics, especially in connection with mixed boundary value problems. Different con- tact problems of theory of elasticity, problems on strained states near cracks, some problems of elec- trostatics and other topics are among them. T h e theory of dual equations recently became very de- veloped and is the subject of numerous investiga- tions. Formal analytical methods for finding solu- tions to dual equations have been studied by many authors (see [4, 9]), b u t much less attention has been paid to solvability question of these equations.
The dual integral equation of Titchmarsh's type was investigated in [12] by distributional approach and in [l| on Lebesgue spaces. T h e solvability of dual integral equations involving Fourier trans- forms and dual series expansions involving orthog- onal expansions of generalized functions was con- sidered in [5-7]. Note t h a t , dual integral equations for convolutions can b e also reduced to t h e equa- tions involving Fourier transforms. Some classes of those equations on semi-axes of t h e real axis were investigated in [3].
To our knowledge, t h e solvability of t h e systems of dual equations has been little studied. T h e aim of the present work is t o consider the existence and uniqueness problems for systems of dual integral equations involving Fourier transforms of general- ized functions, which are a generalization of some systems of dual equations encountered in t h e mixed boundary value problems of m a t h e m a t i c a l physics and contact problems of the elasticity.
2 Functional spaces
We will need t h e following notions in [2,10,11]. Let K be a real axis, S = S(R) a n d 5 ' = 5 ' ( R ) be t h e
^Te!: 097fl79122.S
Schwartz spaces of basic and generalized functions, respectively. Denote by F and F ~ ' the Fourier transform and inverse Fourirr tran.sform defined on S'. It is known t h a t these operators are automor- phisms on S'. For a suitable ordinary function f(x) (for, example, / € L^{R)), the direct and inverse Fourier transforms are defined by
nm) = /"
/ { i ) e " « < i i .A Fourier transform of a generalized function / € S' IS defined by the formula [10]
< F[/],iw > : - < f,F[-ui] >, w€S.
Inverse Fourier transform F~^ of tions is defined by
< f{-x),w(x) > : = < f.w{-x) > , where < f,-w > denotes a value of the generalized function / e 5 ' on a basic function w eS, besides, if,-w) :=< f,W >. Let H^ := H^{R)(s e R) be the Sobolev-Slobodeskii space defined as a closure of t h e set C7^(R) of infinitely differentiable func- tions with a compact support with respect to the norm [2]
Hull. : = [ | " (1 + l ^ l f ^ m O l ' d ^ ] ' ^ ' < oo, ii = F[u]. (2.1) T h e space H^ is Hilbert with the following scalar product
(«,«). -ly^ \(\fu{am)d(, (2.2)
Let S7 be an interval in M. The subspace of i f ^ R ) consisting of functions u(x) with supp u c f! is
Nguyen Thi NgSn Tsp chl KHOA HQC & C 6 N G N G H S 99(11): 169 - 175 tlenoted by H^,{i\)\9], while t h e space of functions P r o p o s i t i o n 2 . 1 . The subspace HfCn) is closed in i'(.r) = p u ( x ) , where u e /f"{R) and p is the re- H""(Ill).
sl.iictiun operator Lo Si is driiofcd by ir{il). T h e norm in //"(17) is defined by
Pmof. Let u S 1H'(R),U,„ £ H ; ( ! ! ) ( m = 1,2,...) ll"llH.(n) = iuf II'"!!., a n d |!u - u,,,!!,- -» 0, m -> oo. We shall show that , „ . „ . . . „ . , , . u € im:(r!). We have
where t h e mnniuni is tHkeii over all possible cxteli-
sions (,. e H"(R). <„,„,,,„j>=o, Vi;, e c r ( R \ n ) .
LcM, A' hi' a linear topological .spiirc We deiiote t h e
(iiiecl product of 2 elements A' hy X ^ . A topology Therefore
in A''^ is f^iveii by t h e usual topology of t h e direct 2 p r n d u r t s We shall use bold letters for denoting < u,„,vir > = 0, Vw € ( c ^ ( R \ n ) j . (2.5)
\'ector-vaIues and matrices. Denote by u a vector of t h e form u = (ti|,Wa), and
For t h e \eclors u e ( S ' ) ^ . w £ (S)^ we p u t
Using t h e Cauchy - Schwartz inequality, from the general Farseval equality for a n y w € ( C ^ ( R ) j ,
< U, W > - ^ < Uj, UJj > .
The Fourier transform a n d t h e inverse Fourier ~ 2^\2^ j l " j U J ~ " m , K J 1 W J K ; W S | transform of a vector u € (S')^ are t h e vectors • ' ^ '
F ^ ' j i i ] = ( F ± ' [ u i ] , F * ' ( u 2 ] ) . defined by t h e equa- ^ tions -
< F i u , ( 0 , w ( ^ ) > = < u ( i ) . F H ( x ) > ,
^ ^ | | u - u ^ | | , | | w | | _ r , V w e ( C - ( R ) ) . (2.6 Prom (2.6) it follows t h a t , if u,„ -> u m H J ( n ) ,
< F - ' [ u ] ( 0 . w { 0 > = — < u ( i ) . F [ w j ( - i ) > , El^e" u ^ -> u also in (S')^- Passing t o limit in (2.5) 2 T we have
< u , w > = 0, Vw e (c„°"(R \ fl)) , IS u e W{n).
P r o p o s i t i o n 2 . 2 . {c^{^)\ w dense tn H f ( n ) . where F -= F^, we S ^
Let / / " ' , Ho'i^), H ' ' J ( n ) be t h e Sobolev- Slobodeskii [2] spaces, where j = 1,2; fi is a certain
sel of intervals in R. We p u t t h a t is u G U'iU). O s = ( s i , 6 2 ) . IHI' = H ' " xH"',
IHlf(Jl) - H^'lil] X H^^Q), H*(ri) = H " ' ( f i ) ^H^^iQ).
A scalar product and a norm in H'" and IHlf(n) are Proof- Since C ^ ( a ) is dense in Ho'{Q) (j given by t h e formulas 1.2) [2], then for a n y e > 0 a n d a given func- tion 1/^ € Ho'i^i) t h e r e exists a certain function V ^ , , I I I , / V I I 1 1 2 ^ ' ' ^ W] € C ^ ( f i ) - such t h a t
fu.v),- = ^ ( U j . U j ) , , , | | u | U - = ( ^ 2 ^ | | u j | | ^ J , J
(2.3) \\^,--^j\U,<-~-
where ||Uj|!sj a n d ( u j , U j ) a j a r e given by t h e formu-
las (2.1) and (2.2) respectively, A norm in W{il) fVom this it follows is defined by t h e equation
i!ui,„.-„„:^(i:ii«.ii«-..,)'" (-) "-""-£"--'"^|i2-'-''<^-
J - 1
where . ? = (»i,S2), » , e R 0 = 1,2). T h e „rn„f is comolete. 0 170
Nguyen Thj NgSn T9p chf KHOA HQC & CONG NGHg
99(11): 169- 175 Theorem 2.3. Let 1H1'''*(R) fte a dual space of thespace 1I*(R). Then ]H1^'*(R) is isomorphic to the space 1H1~''(R). Moreover, a value of a functional f e ]H1~*(R) on an element u G H*(R) is given by the formula
(f,u)„ = x;y°°/,(()«;(()«,
rfcR, ffj(t) = Fluj\{t).f,{t) = Fmt).
(2.7)
Proof. Due to the Riesz theorem on the general form of linear continuous functionals in Hilbert spaces any functional *(u), u € IHl'(R), there exists an element v e H^'(R), such that *(u) = (v,u),- and its norm ||*|| = sup||„,|.^, ||*(u)|| = ||v|l,-.
Denote
f(t) = ((1 + |!|)'""i((), (1 + \t\?"n{t)f. (2.8) We have
l'^{i + \t\)-^''\Mt)\-'dt
= l'^(l-i-\t\)''^\%{t)\'dt{j = l,2).
Therefore f := F ' ^ f ] € H-*"(R), ||f||-f - ||v||? =
||$|| and (v,u)3-= (f,u)o, where
2 yoc
{t)dt.
By this, (2.8) establishes an isomorphism between H''*(R) and H"^(R), besides, a value of a func- tional f e H"^(R) on an element u € H'(R) is given by the formula (2.7). The proof of Theorem
2.3 is complete. D Theorem 2.4. Le( Sl C R, u = (ui.ua)'' e
ff"(ii),f G IHl-^^(Si) and If = ((1/1,^2/2)^ be an extension of f j^m Q to U belonging to 1H1"'(R), then the integrals
Proof. Let I'f be another extension of the f. Then we have If - I'f = 0 on fi, i.e.
(If - I'f, w)„ = 0, Vw € {C^in)f. (2.10) Since (C^iU))^ is dense in lHlf(f2), then from (2.10) It follows
( l f - l ' f , u ) „ - 0 , Vu€lHl*(il), that is (I'f, u)„ = (If, u)o. Thus integrals in (2.9) do not depend on the choice of the extension If. From (2.9) we obtain
|(lf,u)„|«||u|l,H||lf||_.-.
Since (If, u)o does not depend on the choice of If then
|(lf,u)„|<||ul|,inf||lf||_.-=||u||,||f||„-,,n,.
(2.11) Thus, every element f e HI "(Sl) gives a continu- ous functional on Hf(fl) by the formula (2.9). Let
$(u) be a linear continuous functional on Ho(fl).
The space IHlf(Ii) C H*(M) is a Hilbert space with respect to the scalar product (2 3). Therefore, due to Riesz Theorem there exists a function vector v € Hf(Sl), such that *(u) = (v, u)^. We put Ut) - ((1 + |f|)^^'S.{t),(l + \t\)^'^Mt))^, fo = F - ^ i ] - Then f^ € //"""(R), pf^ = f G H-^(fl).
where p denotes the restriction operator to Sl. We have $(u) = (v,u)^-= (fo, u)^ and ||0|| = \\v\\g = l!fo!l-s > llfilH-'iO)- On the other hand, in virtue of (2.11) we have ||*|| - sup||„||^_^i ||*(u)|| <
l|f||H-r,„). Like this, 11*11 = II/IIH-IS.)- The proof
is complete. n
[f,ul: = (lf,u)„ := E / ° ° ljf,{t)<'i',t)dt (2.9)
do not depend on the choice of the ^tension If.
Therefore, this formula defines a linear continuous functional on IHlf(Sl). Conversely, for every linear continuous functional 3'(u) on IHIo(fl) there exists an element f e H"^(il) such that $(u) = (u,f) and||0|| = ||f|lH-.(n).
Pseudodifferential tors
opera-
In this section we shall investigate pseudodifferen- tial operators of the form
lAu){x):=F'^\A{t)n(t}]{x), where A(f) = |la,j(e)||2x2 is a squai'e niiitiix of order 2, u = (ni,U2)^ is a vector, transposed to the line vector (1^1,02)) and u(*) := F[u] ^ (F[ui],F[u2])^- We introduce following classes.
Definition 3.1. Let a € R. We say that the func- tion a(t) belongs to the class (T"(M), if
|a(t)| ^Ci(l-hlfl)'*, VtGR, and belongs to the class o%(R), if
C 2 ( l - H l t | ) " ^ a ( t ) ^ C i ( l - H t | ) ° , VtGR, (3 1) where C^ and Co are some positive constants.
171
NguySn Thj NgSn Tgp chf KHOA HQC & C 6 N G N G H $ 99(11): 1 6 9 - 175 P r o p o s i t i o n 3 . 1 . Let a{t) > 0 and such that
(1 -I- \t\)~°'a(t) is a bounded continuous function on R. Suppose moreover Ihtil, llnrv are the positive limits of the, function {\ I |/|) "all) lohenl -> I'Xi Then a(t) € oliR).
Proof A.suiiiiiie tlint
^_limJl + | ( | ) - " a ( 0 = A±, A ± > 0 . Then for any positive small enough number £• < A+
there is a positive number Ra > 0, .sueh t h a t A_ ^ ( H - j ? | ) - ' ' a ( 0 ^ A+, |t| > / ? „ , where
X- = min{A± - f } , A+ = max{A± + e).
I-\irllier, from the assumption it follows t h a t the function (l + | ( | ) ' ' ' a ( 0 a t t a i n s t h e greatest value M and the smallest value m in tlie interval {-Ro. Ro\- Therefore. (3.1) holds with
Cl = m a x { A + , M } . C2 = m i n { A _ , m } .
This completes the proof D D e f i n i t i o n 3 . 2 . Let A(f) = j|a,j(<)ll2x2, t € R
be a square matrix of order 2. Assume t h a t , a,j(t) are continuous functions on R. f>j G R, ( j = 1,2).
Denote hv Q — ( 0 1 , 0 2 ) ^ and by E ° ( R ) the class of matrices A{t} = ||a,j(()||2x2. such t h a t
a „ ( f ) e a < ^ ' ( M ) , a,,{t)€a^"iR), Pa ^\ia,+ctj).
(3.2) We shall say t h a t the matrix A ( ( ) belongs ti> the class E ° ( R ) , if A ( 0 is Hermitian, i.e ( A ( 0 ) ' = A{t}. and satisfies the condition
2
w ^ A w > C, J ^ ( l - 1 - [ t D ^ J | u ' J | ^
Vw = ( ^ « I , u ; 2 ) ^ 6 C ^ (3.3) where Ci is a positive constant a n d w is t h e com- plex conjugate of w. Finally, t h e matrbc A ( 0 be- longs to class E ^ ( R ) , if
R e w ^ A w > 0, Vw = (^1,1^2)^ € C ^ (3.4) P r o p o s i t i o n 3 . 2 . Let the matrix A ( t ) = A+(*) belong to the class E ^ ( R ) . Then scalar product and norm in 1H1'*^^(R) can be defined by the foi-mulas
( u , V ) A , . a / 2 = / " F p l ( * ) A + ( t ) F [ u l ( / ) d f , (3-5)
a
0 0 •. l yF | u ' - K ( ) A + ( ( ) F l u | ( ( ) « ) respectively.
{3.ej
Proof Using t h e Cauchy-Schwartz inequality one can show t h a t
w ( ( ) ' A w ( / ) ^ C a X ^ i l + | t | ) " ^ K ( t ) l ' . {3-7) j = i
where Ci is a positive c o n s t a n t . Replacing in (3.2) and (3,7) w,(t) by Uj(() = F\uj]{t) a n d w ( i ) by F [ u ] ( i ) , and after t h a t , integrating on {-^.-yi).
we have
Cijzj^ (i + \t\r'\F\u,m'dt
i j ^ F | u ' - ) ( ( ) A + ( ( ) f | u ] ( ( ) > i (
^ c , Y . j ( i + ! i | ) " ' ! f K K ' ) l ' * - (3.8)
n-om (3.8) due t o (2.1) a n d (2.3) follows (3.6). It is clear t h a t , t h e integral (3.5) defines a scalar prod-
uct in 1H'"''(R). D
T h e following proposition can b e easily proved:
P r o p o s i t i o n 3 . 3 . Let A ( ( ) € E ° ( R ) , u e jjn/Q^mj J•j^f^ ^ g F o u r i e r integml operator.Au de- fined by the formula
{Au){x):=F-'{A(t)u(t)\{x) (3.9) IS bounded from E^I'^{R) into H - " / 2 ( R ) .
P r o p o s i t i o n 3 . 4 . Let Sl 6e a bounded subset of intervals m R. Then immersion of H^(Sl) into H^"^(S1) IS completely continuous, where £ — i£u62)>0{<^£j> 0,3 = 1,2).
Proof. T h e proof is based on t h e fact t h a t the im- mersion J/''J(S1) into H^'~^'{Q),£j > 0 is com- pletely continuous if Sl is bounded in R, j — 1,2 (see [9, 10]).
Nguyin Thi NgSn Tap chl KHOA HOC & CONO NGHE 99(11): 169- 175
4 Solvability of s y s t e m s of dual e q u a t i o n s
In this section we shall investigate a system of dual equations in t h e form
|pf-'[A(aa(5)](x) = f(i), l e s i . yf-'ia(0](l) = g(x), ie!l':=R\!!,
(4.1) where Sl is a certain interval in R, ii(?) = (ui(5),U2(0)^ is a function vector t o be found,
fix) = ifi{x),f2{^))^ e (I''(S1))2. g(x) =
(ffi(a^).33(3:))^ € (X>'(S1'))^ a r e given vectors of distributions on fi and fl' respectively, A(^) =
||Q,J(^)||2>;2 ly a given square matrix a n d is called the symbol of the system (4.1); p a n d p ' are re- striction operators t o fl and Sl' res])e(t,i\ely T h e operator F~ ^ is understood in t h e sense of gener- alized functions.
We shall consider t h e system of dual integral equa- tions (4.1) under t h e following conditions
f A ( O G S f ( R ) . a n d A ( ^ )
< is positive-definite for almost ? G R, (4.2) [{(x) € E - ^ / 2 ( n ) , g ( x ) G H ^ / 2 ( n ' )
and shall find t h e vector u ( ^ ) in t h e form u(^) = F[u](0, where u e H ^ ^ 2 ( R ) .
T h e o r e m 4 . 1 . (Uniqueness) Under the assump- tions (4.2), the system of dual equations (4.1) has at most one solution u{x) - F - ' [ u ] ( 2 ; ) e H^/=(R).
Proof. To prove t h e theorem it suffices to show t h a t the homogeneous system of dual equations
fpF-'[A(0i3(01(x) = 0, x e n ,
\ p ' u ( i ) = p ' F - ' [ a ( ? ) ] ( x ) = 0, X e f!' := R \ fJ, has only the trivial solution.
Since u e ]Hlo^^(n), t h e last system may be rewrit- ten as
(ylu)(x) = 0, X e fi, (4.3) where
( A u ) ( i ) = p F - ' ( A ( O a ( 0 ] W , x s n . (4.4) Since Au s H - ° ' ' ( n ) ~ ( B l f ' " ( ! i ) ) ' (see Theorem 2.4), from (2.9) we have
[Au,u\= f" a7\i).F{lpF-^lA&]){0d(.
Since t h e last integral does not depend upon a choice of / p F " ^[Au] we can take the extension in the form
/ p F - ^ [ A u ] - F - ' [ A u ] . Hence we have
[An, u] = 1^ WW)-MaM^)d^- (4.5)
TIKMI from (4.3)and (4.5) we get [Au,u] = p W{X).A{S,).n{e.)d£, ^ 0.
Since u ^ ( 0 - A ( 0 - u ( 0 > 0, from this it follows t h a t 11(0 = 0 , u { x ) = 0 (see (3.4), (4.2)). T h e proof of
Theorem 4.1 is complete. D L e m m a 4 . 2 . The system of dual integral equations
(-1.1) is equivalent to the following system of dual equation,s
p F - ' | A ( O v ( 0 ] ( a : ) = (ix) -pF-'lAiOViimx),
xesi.
(4.6) where v - F - ' [ v ] G H?^^{Si) satisfies the lelation v + l'g = u G e " / ^ ( M ) (4 7) (I'g 15 an arbitrary extension of the generalized vector-function g from i l ' into U).
Proof Assume t h a t u G 1H1'^''^(R) satisfies the sys- tem of dual equations (4.1) and I'g G 1H1°''2(R) is an arbitrary extension of g € EI"^^(S1'). Taking V = u - I'g, we get v G Mt^^(U). Putting (4.7) into (4.1) we have (4.6). T h e right-hand side of (4.6) belongs to lHl~^/^(fl) in view of Proposition 3.3.
Conversely, assume t h a t v G Mo^^{Q,) satisfies the equation (4.6). Then obviously, t h e vector-function u defined by (4.7) belongs t o 1H1'5(R). We shall prove that this function satisfies t h e system of dual equa- tions (4.1) in t h e sense of distributions. Indeed, by transfering t h e second term in t h e right-hand side of (4.6) to t h e left-hand side and using (4.7) we ob- tain t h e first equality in (4.1). Further, from (4.7) it follows t h e second equality of this system of duaV
equations. ^ Denotes
h(x) = f(x)-pf-'|A({)l'g(01(x).
(4.8) 173Nguyin Thi NgSn
Tjp chl KHOA HOC & C 6 N Q N 0 H $99(11): 169- 175 Using (l.'l), (4.8) we rewrite (4.0) in the form
( / l v ) ( x ) = li(..). ! € ! ! . (4.9) Our purpiLse now is t o establish t h e existence of the solution of t h e svslein (4.9) in t h e space lHl"^^(fi).
We shall consider t h e following cases.
4 . 1 T h e I A(?)e5:?(R)
In this ease due to Proposition .').'2 t h e seiilar prod- uct a n d norm in 1H1"^^(R) can be defined by t h e formulas (3.5) a n d (3.6):
( V . W ) A , , S / J . f " TW]U}^ilt)Flv]{t)dt,
l | v i ; A , . « / 2 = ( y ° ° ? P K 0 A + ( ( ) F | v | ( 0 . « ) ' " ,
respectively We shall also write A + v instead of -IvT h e o r e m 4 . 3 . (Existence) / / h G M-"'-{il). A ( 0 = A + ( i ) G i:^, then the system (4.9) has a unique solution v G Ho (fl)- Proof. By an argument similar t o t h a t used in t h e proof of the Theorem 2.4 we have
L 4 . v , w ; ^ y " F\wT\(t)A^it)F\v]it)dt
= ( v , w ) A ^ . a / 2
for arbitrary vector-functions v and w belonging t o H„ (Sl). where [.4+v, wj is defined by t h e formula (2.9). Therefore, if v G H^^^(fl) sati.sfie.s (4 9). then the following equality holds
( V , W ) A ^ . S / 2 = [ h , w ] , Vw € Hf/2(S1). (4.10) We shall d e m o n s t r a t e t h a t , if (4.10) holds for any w G H ° ( n ) , then t h e vector-function v will satisfy the system of equations (4.9) in t h e sense of gen- eralized functions on Si. In fact, noting t h a t (4.10) holds for w = ^ G ( C ^ ( n ) ) ^ a n d using t h e formula [10]
/ 7 ( 0 ^ d ^ = 2 7 r < / , i ? > = 2 7 r ( / , v . ) . we get
[ h . v ' ] = y " l h ( 0 ^ d e = 27r{lh,v),
(V,<^)A^.5/2 = J " F[(^^](OA+(OF[v]{e)de - 2 7 r ( F - ' l A + v ] , ( ^ ) .
Hence, from (4.10) it follows
iF-'[A+v],<p) = ( I h , v ) , V ^ G ( C r ( i ^ ) ) ' , i.e.
p F - ' [ A + ^ ] ( 3 : ) = 7;lh(i) = h ( i ) . x € U.
We now return t o t h e relation (4,10), Since [h, w] is a linear continuous functional o n t h e Hilbert space Ho (Sl). then by virtue of t h e Riesz theorem there exists a unique element v„ e H o ' (Sl). such that
[h.w] = ( V O , W ) A ^ . A / 2 , W G lHI^''^(!i).
a n d moreover, there is t h e estimation
I|V„!IA».„-/J < C|!h|!„-.„,„,. (4.11) where C is a positive c o n s t a n t .
Since (4.10) is equivalent t o (4.9), t h e system (4.9) has a unique solution v — VQ G Ho (Sl), a n d this completes t h e proof of T h e o r e m 4.3. Ci R e m a r k 4 . 4 . I t is easily t o see t h a t t h e inverse op- erator A + ' from 1HI"°(S1) into H°(S1) is bounded.
This follows from Theorem 4.3 and t h e inequality (4.11).
R e m a r k 4 . 5 . T h e solution u of t h e system of the d u a l integral e q u a t i o n s (4 1) expressed in terms of the solution v of t h e system (4.9) by t h e formula (4.7) does n o t depend o n a choice of t h e extension ig-
T h i s fact follows from t h e uniqueness of t h e solu- tion of the s y s t e m of d u a l e q u a t i o n (4.1). Hence we can choose a n extension Ig such t h a t
l|lgilA^.o/2^2||g||H.-/«,n-)-
In this case, from (4.7), (4 8) a n d (4.11) it is easy to obtain t h e following e s t i m a t eI!U!IA».»-/2 < C ( | ! f | | „ - . , . , „ , -I- !|g!l„„-,.,„,,), (4.12) where C = const > 0 Therefore, t h e solution of t h e system of d u a l e q u a t i o n s (4.1) depends con- tinuously upon t h e vector-functions given on the right-hand side of this system. Hence, t h e follow- ing theorem h a s been proved.
T h e o r e m 4 . 6 . (Existence) Let A ( 0 ^ E J ^ ^ ( R ) , f G M-°f^{il), e G m-^l^{Sl'). Then, the system of dual integral equations (4.1) has a unique solution u = F ~ ^ [ u ] G IHI"^^(R) and satis- fies the estim,ation (4.12).
Nguyen Thi NgSn T9P chi KHOA HQC & CONG NGH$ 99(11): 169-175
4.2 The case A(^) G Sf (E)
We assume in addition t h a t t h e set fl is bounded and there exists a square m a t r i x A + ( ^ ) G S + ( R ) ,
B ( ^ ) : = A ( C - A + ( a G E ' ^ - ' ' ( R ) , (4.13) where 0 = (PuPi)"^ G R ^ /3j > 0 {j = 1,2). Now we represent the operator A defined by (4.4) in the form A — A+ + B, where
Proof According to Proposition 3.2 the system of dual equations (4.1) is equivalent to the system (4,9). In virtue of Remark 4.4, the operator A:^^
is bounded from H - " / 2 ( n ) into Hf^^(Sl) and due to of Proposition 3.4, the operator B v defined by (4,14) is completely continuous from Hf'^^(Sl) into E'"^'^{ii). In this case, we represent the system (4,9) in the form
A + v - | - 5 v = - h . Hence, we have
A+v - p F - ^ [ A + v ] , Dv = p F - ' [ B v ] ,
(4,14) v-\-A+^Bv = A+^h. (4 15)
T h e o r e m 4 . 7 . (Existence) Suppose that the set n is bounded. Under conditions (4,2) and (4.13), for every f G H - ° / = ( S l ) , g G H^/=(S1') the sys- tem of dual equations (4.1) has a unique solution U = F - H U ] G H ^ / 2 ( R ) .
Since the operator A^^B is completely continuous, the system (4,15) is Predholm, and from the unique- ness of solution it follows t h a t this system has a unique solution. Therefore, in this case, the system (4.1) has a unique solution u G ]HI"/^(H). The proof
is complete. Q TINH GIAI DUDC CUA HE PHUDNG TRINH CAP TICH PHAN v D l PHEP BIEN DOI FOURIER
Tom tat. Trong bai bio nSy chung tfli xet linh giai duoc cua he phitong trinh cap tfch phan Fourier tdng qual. Da chiing minh cac dinh ly t6n tai vS duy nha't nghifim ciia he trong kh6ng gian Sobolev ciia cic ham suy rfmg. ChiJng minh tfnh t6n tai va duy nhft nghidm trong cac trucmg hop bi^u tnmg A ( f ) G E f (R) va A(^) G Df (M).
Tir k h o a : Bien ddi Fourier, he phuong trinh cap Ifch phan, bai toan bien hOn hop.
References
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