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TWO-DIMENSIONAL SINGULAR INTEGRAL EQUATIONS OF THE HYPERCOMPLEX FUNCTIONS
1996,16(1):44-50
TWO-DIMENSIONAL SINGULAR INTEGRAL EQUATIONS OF THE HYPERCOMPLEX FUNCTIONS 1 u«
Chungen ( :*'Jj.~ ) Dept. oj Math., Nankai Unw., Tianjin 30071, China. Abstract In this paper, it is considered for some two- dimensional singular integral equations of the hypercomplex functions in the Douglis sense. In some special cases, the Fredholm, conditions and index formula of such equations are obtained. Key words
Singular integral equation, Hypercomplex function, Index formula.
1 Introduction Let D := 8z+q(z)8z be a Dauglis operator in the space of hypercomplex functions C 1(G), where the hypercomplex function q(z) is nipotent and belonging to BO,a(c), 0 < a < 1~ and let t( z) be a generating solution of the operator D, namely, t( z) = z + T( z) and Dt( z) = 0 in C and T(z) E B 1(C). where G is a bounded domain enclosed by a finite number of C 2 curves. The following integral equations are considered
a(z)w(z)
-
+ b(z)w(z) + c(z)
+d(Z)!
I lG
~
[t(() - t(z)]2
! lG(
dG(
w(()
[t«() _ t(z)J2dG(
+ Kw(z) = h(z),z E G,
(1)
where all a(z), b(z),c(z), d(z), h(z) are the hypercomplex functions, w(z) is unknown hypercomplex function belonging to £1'( G), p > 2.G is a bounded domain with a smooth boundary in the plane. Let us now assume that a(z), b(z),c(z), d(z) E C(G) n Wf( G), h(z) E LP(G),p > 2. Let fee) = ~, then from (1) we obtain
t;Dtm
al(z)!(Z) +d(z)! 1 Received
-
!
f + b1(z)!(z) + c(z) lG
I t,DDt«()!«()dG, JG [t«() - t(z)]2
Nov.2, 1993; revised Jul.13,1994.
t(Dt(()f(() [t«() _ t(z)]2 dG(
+ Kd(z)
=h(z),z E G
(2)
Liu: TWO-DIMENSIONAL SINGULAR INTEGRAL EQUA.TIONS
No.1
45
where Ql(Z) = t,Dt«()a(z), b1(z) = t,Dt«()b(z). Let
T f=-!.J f t(Dt(()f(()dG G 1r JG t«()-t(z) "
n: f = _!. J f t(D@f«() dG G 11" JG [tee) - t(z)]2 " nGf =nGf - uf, a
= tt rz ,
(3)
(4)
the equations (2) have the following form
A(z)f(z) + B(z)f(z) + C(z)nG/(z) + D(z)nGf(z) + Kf(z) = g(z),
.
(5)
where A(z) = al(z) - 1I"c(z)0", B(z) = b1 (z) - 1rd(z)a-, C(z) = -1rc(z), D(z) = -rd(z), A(z), B(z), C(z), D(z) E C(O) n Wf(G), (p > 2).
2 Two Special Cases Before all, we consider the following case
fez) - b(z)IIGf + K f = g(z),
(6)
where fez) E V(G) is a unknown hypercomple.x function, g(z) E IJ'(G),p > 2, and b(z) are hypercomplex functions, and Ib(z)1 :5 bo < 1, K is a integral compact operator in the space Lp(G). By [4], we know that IlII G IIL2 = 1, then exist some e > 0 such that if 2 < p < 2 + e, we have
IIb(z)IIGIlL p < 1 and the inverse operator (I - b(z)IIG)-1 exist, and it is a bounded linear operator also. From (6) we obtain
fez) + Kif = 91(Z).
=
where K 1 (I - b(z)IIG)-lK, is a compact operator in IJ'«(;).91 = (I - b(z)II G ) - l g E £P( 0). From above discussion we have proved following result Theorem 1 The equation (6) has unique solution for every g(z) E £P(G), or its homogeneous equation has non-zero solutions. Besides (6) we can also consider following equation
fez) - b(z)IIGI + KI = g(z),
(7)
where Ib(z)1 :5 bo < 1. Belonging to the space V(O), we can obtain the same result for this case as the case (6). We still consider the special case in the equation (5) with the conditions B(z) = C(z) 0, namely we consider the following equation
=
A(z)f + D(z)UGf = g(z).
(8)
46
VoL1S
A.CTA MATBEMATICA SCIENTIA.
If fez) E V(G),p
> 2, is a solution of equation (8) then the function w(z)
= TGI =
_.!.
1r
JJGI
t(Dt(()/(() dG t«() -t(z) ,
is a hypercomplex continuous function in the whole plane C, and is hypercomplex analytic in the domain C \ G (let us define G C \ G) and vanished at infinity, and = I, = ITG f. where 8, 8 is defined just as [4], therefore we obtain
aw
=
. A(z)8(w) + D(z)8w = g(z), namely, we have
8(Aw + Dw) - (8A)w - (8D)w
oW
(9)
= g(z).
Let us define
cp(z) = A(z)w(z) + D(z)w(z).
(10)
then cp(z) = D(z)w(z)+A(z)w(z), if A(z)A(z)-D(z)D(z) is invertible, namely, the complex part of A(z)A(z) - D(z)D(z) is not vanished, let us define cpt f be the complex part of the hypercomplex function f, i.e. cpt I = 10,(1 10 + Ile l + ... + In_len-I), then the condition is IcptA(z)1 ;/; IcptD(z)l. In view of (10),
=
w(z)
=
1 A(z)A(z) _ D(z)D(z) (Alp - DIj),
then we obtain that
(11) where
AO(z) = i(iaA - baD), BO(z) = t(i8D -baA), ~
= A(z)A(z) -
D(z)D(z).
Let
~
ep(z) = { Ip(z),z E w(z),z E G.
(12)
then ep(z) satisfies the equation (11) in the domain G. and in the domain Gis a hypercomplex analytic function and vanished at infinity. In view of the continuity of the function w(z), we obtain (13)
l1(Z) = {
leA¥' - DIj),
z
~(z),
%
-
e~
E G.
Liu: TWO·DIMENSIONA.L SINGULAR INTEGRAL EQUA.T10NS
No.1
=
=
47
=
we obtain A(z)811 + D(z)8v g(z), let p 811, then 11 +O(z) + TGP, where .0(%) is a hypercomplex analytic function in G, because of 11 and TGP are hyperco mplex analytic functions in G. as a result, ~O(z) .can extend through to the whole plane C, such that it is a hypercomplex analytic function vanished at infinity. In view of the Liouville's Theorem, we have ~O(z) == 0, thus 11 = TGP. from that p(z) will be a solution of equation (8). On other hand, we define
Ao(z)
Bo(z) 90(Z)
={
A
0(%) Z E G,
O
ZEG;
= {BO(Z)
ZE
o
z E
~l
q;
~l
= {g(Z)
ZE z eo.
o
then q,(z) satisfies the following equation in the "hole plane C
fJ
(14)
11, 12, · .. , lie
are the basic of the solution space of homogeneous equation (B).Let 'PI, ~2, •.., «Jk are corresponding solutions of adjoint problem (13) and (14) respectively by above Let
means.
If
Cl, C2,· •• , Ck,
and
k
E Cjipj(z)
j=1
= 0, z E C. In view of (10) and (12), then
k
E CjWj(z) = 0, from this
j=1
take fJ to the two sides of above equation, then
It
E Cj Ij = 0 => CI =C2 = · · · = Cit = o. From
j=l
this we have obtained that the dimension le' of the solution space of the adjoint problem (13) and (14) satisfying le' ~ k: Inversely, Let ~l, ~2,··· ,~k" is the basis of the solution space of the adjoint problem (13) and (14), and PI, P2, ••• , Pic' are the solution of the equation (8) made by the following method
If
k'
Ie'
Ie'
Ie'
j=1
j=1
j=1
i'
E CjPj = 0 => E Cjl1j = TG(E CjPj) = 0 ~ E Cj+j .
j=1
=0 ~
Cj
= 0,;
:s k, from above we have shown that I: = k'. So, we have proved following Lemma 1 If IcptA(z)1
= 1,···,~1 =>
:F IcptD(z)l, then the equation (8) is equivalent to the adjoint
problem (13) and (14), and the dimension of the solution space of the homogeneous equatioo
48
ACTA MATREMATICA SCIENTIA.
Vol.16
(8) equal to the dimension of the solution space of the homogeneous adjoint problem (13)
and (14). Let ¥>o is a solution of the homogeneous equation ·(14), then
8cpo - Ao(z )~o - B o( z )<,00 = 0 we consider
Wo(z) =
where
g(z)
_! ,..
Jf
t,Dt(()g(() dG lG g«) - t(%) "
={ Ao(z) + Bo(z)::~:i, Ao(z)
(15)
+ Bo(z),
/po(z) # 0, cpo(z) = o.
Defining iJo(z) = cpo(z)e-U7 O(z ) , then
8~o(z) = 8/po(z)e-" o(z ) + /Po(z)( -Ao(z) - Bo(z) ::f:~ )e-tDo(z)
=(8cpo(z) -
Ao(z)cpo(z) - Bo(z)cpo(z»e-wo(z) = o.
Therefore, iJo(z) is a hypercomplex analytic function, as a result, every solution of the equation (15) has the following form (16)
where iJ(z) is a hypercomplex analytic function. w(z) is a continuous in the whole plane and hypercomplex analytic in (; and vanished at infinity. In view of following formula if 1 U E C (G ), then -
the
equ~tioD (14)
J1 G
t(Dt«)8u dG (zoE G t( () - t (Zo) ' ,
(17)
equivalent to following Fredholm integral equation
lfJo (%) where
1
1 u(z)dt(z) - 1 21ri r t( z) - t (zo) 1r
( ) -UZo
-!
11"
Jf
lG
91(Z) = _1
t,Dt(()rAo(()/po(() + Bo«()~ dG tee) _ t(z)
f
lfJo«()dt«) _
21ri Jr t«() - t(z)
.!. 11"
JJGf
, - 91
t,Dt(()go(() dG t«) - t(z) (.
() Z ,
(18)
(19)
Let us show that homogeneous equation (18) only have the trival solutuion. Let CPo be such asolution, then lfJo(z) satisfies the equation (15), therefore,
Liu: TWO-DIMENSIONAL SINGULAR INTEGRAL EQUATIONS
No.1
where
r1(
Fron~
49
, z ), r 2 « , z), are resolvent kernel (see [1]1. (19) and (20), we obtain
= 2~i Jr{}(l)(z,()cR()dt«) -
-~
J fG[{1(1)(z, ()go«) -
10(1)(Z,() 10(2)(z, (),
n(2)(z,()C)«)dt«)
0(2)(z, ()po«)]dG(
t(()~t( .. ) 1= o(lz - (1- 2/P)
= o(lz -
(J-2/ p )
From (14) and note that tpo is hypercomplex analytic in we obtain that
where 0(1)(Z, ()
•
O(2)(Z, ~)
O(l)(Z,
={ =
1
o
and vanished at infinity,
z E G,
t(,)-t(z)
{{1(2)(Z,
G,
Z
o
E
G;
z E G,
_
.0
z E G;
g*(z) = { ; : f fG[O(l)(Z,()g«() - 0(2)(z, ()g«()]dG"
z E G, z'E
G.
By the Plemelj-Soknotki's formula (see [2]), we have 1
1
f
C)«)dt«)
= 21l"i Jr t«() _ t(z) -
1
2~(z),z
E f.
As so, because of (13), we obtain following singular integral equation.
1 2(1 + A(z»~(z)
_ A(z) 211"i
f
1 1 f () + 2D(z)~(z) + 21ri J 0 1 (z, ()~«()dt«()
~«()dt«() t(z)
Jr t«) -
1 --2. f 1r~ Jr
r
+ D(z) f 21ri
iWdt(() t(z)
Jr t«) -
0(2)(z,()~«()dt«() = g*(z)
(22)
Take conjugate to (22) we obtain 1
--
1--
1
f
-
2(1 + A(z»~(z) + 2D(z)~(z) - 21l"i Jr 0(1)(z, ()~«()dt«()
50
ACTA MATHEMATICA SCIENTIA
A(z) [iR)d@ D(z) +211"i Jr t(() - t(Z) - 211"i
11
+-2. 11"1,
r
Vol. 16
f ~«()dt«() Jr «o - t(z)
-
0(2)(z, ')~«()dt(') = g*(z).
(23)
Let 'It = (~, i), we obtain the character of the equation system (22) and (23) as following
A(z)'1'(z)
B(z)
+ -11"1,.
Ir t (() - t () = r
'It«()dt«() z
*-
-(g (z),g"(z»,
(24)
where
A(z):=! ( l+A(z) 2 D(z)
B(z):=!
1- A(z)
__
( -D(z)
D(z) ) 1+-:1(z) , D(z) ) -l+A(z) ,
and det(A + B) = A(z), det(A(z) - B(z» = A(z). Therefore, the regular condition of the equation system (22) and (23) is that A(z), z E r, is invertible (see [3]) i.e. cptA(z)
1= 0, z
E
r.
(25)
If this condition satisfied, the index of the equation system (22) and (23) is IC
:=
!ind(A(z»r := ![arg(cPtA(z»]r, 11"
11"
therefore, we have proved following theorems Theorem 2 If cptA(z) -# 0, and IcptA(z)1 -# IcptD(z)l, then the adjoint problem (13) and (14) is Fredholm (Noether) problem, i.e. the homogeneous problem (13) and (14) has finite linear independent solutions and the solvable conditions of the inhomogeneous problem (13) and (14) are finite. Theorem 3 If cptA(z) #. 0, and IcptA(z)1 1= IcptD(z)l, then the equation (Sr is a Fredholm equation. Acknowledgment I thank my advisor Prof. Zhao Zhen for his constant support and help over these years. References 1 Gilbert R P, Hile G. Generalized Hypercomplex F\mction Theory. Transactions of the Americal. Mathmatical Society,1974,195. 2 Gilbert R P, Buchanan J L. First Order Elliptic System, A function theoretic approach.New York:Academic Press,1983. 3 Liu Chungen. On Riemann Boundary Problem System of. the Hypercomplex Function.Acta Scientiarum Naturalium Universitatis Nan Kaiensis, 1993,3. 4 SiXWl Huang.Ke Xue Tong Bao (Chinese),1986,'T.
TWO-DIMENSIONAL SINGULAR INTEGRAL EQUATIONS OF THE HYPERCOMPLEX FUNCTIONS 1 u«
Chungen ( :*'Jj.~ ) Dept. oj Math., Nankai Unw., Tianjin 30071, China. Abstract In this paper, it is considered for some two- dimensional singular integral equations of the hypercomplex functions in the Douglis sense. In some special cases, the Fredholm, conditions and index formula of such equations are obtained. Key words
Singular integral equation, Hypercomplex function, Index formula.
1 Introduction Let D := 8z+q(z)8z be a Dauglis operator in the space of hypercomplex functions C 1(G), where the hypercomplex function q(z) is nipotent and belonging to BO,a(c), 0 < a < 1~ and let t( z) be a generating solution of the operator D, namely, t( z) = z + T( z) and Dt( z) = 0 in C and T(z) E B 1(C). where G is a bounded domain enclosed by a finite number of C 2 curves. The following integral equations are considered
a(z)w(z)
-
+ b(z)w(z) + c(z)
+d(Z)!
I lG
~
[t(() - t(z)]2
! lG(
dG(
w(()
[t«() _ t(z)J2dG(
+ Kw(z) = h(z),z E G,
(1)
where all a(z), b(z),c(z), d(z), h(z) are the hypercomplex functions, w(z) is unknown hypercomplex function belonging to £1'( G), p > 2.G is a bounded domain with a smooth boundary in the plane. Let us now assume that a(z), b(z),c(z), d(z) E C(G) n Wf( G), h(z) E LP(G),p > 2. Let fee) = ~, then from (1) we obtain
t;Dtm
al(z)!(Z) +d(z)! 1 Received
-
!
f + b1(z)!(z) + c(z) lG
I t,DDt«()!«()dG, JG [t«() - t(z)]2
Nov.2, 1993; revised Jul.13,1994.
t(Dt(()f(() [t«() _ t(z)]2 dG(
+ Kd(z)
=h(z),z E G
(2)
Liu: TWO-DIMENSIONAL SINGULAR INTEGRAL EQUA.TIONS
No.1
45
where Ql(Z) = t,Dt«()a(z), b1(z) = t,Dt«()b(z). Let
T f=-!.J f t(Dt(()f(()dG G 1r JG t«()-t(z) "
n: f = _!. J f t(D@f«() dG G 11" JG [tee) - t(z)]2 " nGf =nGf - uf, a
= tt rz ,
(3)
(4)
the equations (2) have the following form
A(z)f(z) + B(z)f(z) + C(z)nG/(z) + D(z)nGf(z) + Kf(z) = g(z),
.
(5)
where A(z) = al(z) - 1I"c(z)0", B(z) = b1 (z) - 1rd(z)a-, C(z) = -1rc(z), D(z) = -rd(z), A(z), B(z), C(z), D(z) E C(O) n Wf(G), (p > 2).
2 Two Special Cases Before all, we consider the following case
fez) - b(z)IIGf + K f = g(z),
(6)
where fez) E V(G) is a unknown hypercomple.x function, g(z) E IJ'(G),p > 2, and b(z) are hypercomplex functions, and Ib(z)1 :5 bo < 1, K is a integral compact operator in the space Lp(G). By [4], we know that IlII G IIL2 = 1, then exist some e > 0 such that if 2 < p < 2 + e, we have
IIb(z)IIGIlL p < 1 and the inverse operator (I - b(z)IIG)-1 exist, and it is a bounded linear operator also. From (6) we obtain
fez) + Kif = 91(Z).
=
where K 1 (I - b(z)IIG)-lK, is a compact operator in IJ'«(;).91 = (I - b(z)II G ) - l g E £P( 0). From above discussion we have proved following result Theorem 1 The equation (6) has unique solution for every g(z) E £P(G), or its homogeneous equation has non-zero solutions. Besides (6) we can also consider following equation
fez) - b(z)IIGI + KI = g(z),
(7)
where Ib(z)1 :5 bo < 1. Belonging to the space V(O), we can obtain the same result for this case as the case (6). We still consider the special case in the equation (5) with the conditions B(z) = C(z) 0, namely we consider the following equation
=
A(z)f + D(z)UGf = g(z).
(8)
46
VoL1S
A.CTA MATBEMATICA SCIENTIA.
If fez) E V(G),p
> 2, is a solution of equation (8) then the function w(z)
= TGI =
_.!.
1r
JJGI
t(Dt(()/(() dG t«() -t(z) ,
is a hypercomplex continuous function in the whole plane C, and is hypercomplex analytic in the domain C \ G (let us define G C \ G) and vanished at infinity, and = I, = ITG f. where 8, 8 is defined just as [4], therefore we obtain
aw
=
. A(z)8(w) + D(z)8w = g(z), namely, we have
8(Aw + Dw) - (8A)w - (8D)w
oW
(9)
= g(z).
Let us define
cp(z) = A(z)w(z) + D(z)w(z).
(10)
then cp(z) = D(z)w(z)+A(z)w(z), if A(z)A(z)-D(z)D(z) is invertible, namely, the complex part of A(z)A(z) - D(z)D(z) is not vanished, let us define cpt f be the complex part of the hypercomplex function f, i.e. cpt I = 10,(1 10 + Ile l + ... + In_len-I), then the condition is IcptA(z)1 ;/; IcptD(z)l. In view of (10),
=
w(z)
=
1 A(z)A(z) _ D(z)D(z) (Alp - DIj),
then we obtain that
(11) where
AO(z) = i(iaA - baD), BO(z) = t(i8D -baA), ~
= A(z)A(z) -
D(z)D(z).
Let
~
ep(z) = { Ip(z),z E w(z),z E G.
(12)
then ep(z) satisfies the equation (11) in the domain G. and in the domain Gis a hypercomplex analytic function and vanished at infinity. In view of the continuity of the function w(z), we obtain (13)
l1(Z) = {
leA¥' - DIj),
z
~(z),
%
-
e~
E G.
Liu: TWO·DIMENSIONA.L SINGULAR INTEGRAL EQUA.T10NS
No.1
=
=
47
=
we obtain A(z)811 + D(z)8v g(z), let p 811, then 11 +O(z) + TGP, where .0(%) is a hypercomplex analytic function in G, because of 11 and TGP are hyperco mplex analytic functions in G. as a result, ~O(z) .can extend through to the whole plane C, such that it is a hypercomplex analytic function vanished at infinity. In view of the Liouville's Theorem, we have ~O(z) == 0, thus 11 = TGP. from that p(z) will be a solution of equation (8). On other hand, we define
Ao(z)
Bo(z) 90(Z)
={
A
0(%) Z E G,
O
ZEG;
= {BO(Z)
ZE
o
z E
~l
q;
~l
= {g(Z)
ZE z eo.
o
then q,(z) satisfies the following equation in the "hole plane C
fJ
(14)
11, 12, · .. , lie
are the basic of the solution space of homogeneous equation (B).Let 'PI, ~2, •.., «Jk are corresponding solutions of adjoint problem (13) and (14) respectively by above Let
means.
If
Cl, C2,· •• , Ck,
and
k
E Cjipj(z)
j=1
= 0, z E C. In view of (10) and (12), then
k
E CjWj(z) = 0, from this
j=1
take fJ to the two sides of above equation, then
It
E Cj Ij = 0 => CI =C2 = · · · = Cit = o. From
j=l
this we have obtained that the dimension le' of the solution space of the adjoint problem (13) and (14) satisfying le' ~ k: Inversely, Let ~l, ~2,··· ,~k" is the basis of the solution space of the adjoint problem (13) and (14), and PI, P2, ••• , Pic' are the solution of the equation (8) made by the following method
If
k'
Ie'
Ie'
Ie'
j=1
j=1
j=1
i'
E CjPj = 0 => E Cjl1j = TG(E CjPj) = 0 ~ E Cj+j .
j=1
=0 ~
Cj
= 0,;
:s k, from above we have shown that I: = k'. So, we have proved following Lemma 1 If IcptA(z)1
= 1,···,~1 =>
:F IcptD(z)l, then the equation (8) is equivalent to the adjoint
problem (13) and (14), and the dimension of the solution space of the homogeneous equatioo
48
ACTA MATREMATICA SCIENTIA.
Vol.16
(8) equal to the dimension of the solution space of the homogeneous adjoint problem (13)
and (14). Let ¥>o is a solution of the homogeneous equation ·(14), then
8cpo - Ao(z )~o - B o( z )<,00 = 0 we consider
Wo(z) =
where
g(z)
_! ,..
Jf
t,Dt(()g(() dG lG g«) - t(%) "
={ Ao(z) + Bo(z)::~:i, Ao(z)
(15)
+ Bo(z),
/po(z) # 0, cpo(z) = o.
Defining iJo(z) = cpo(z)e-U7 O(z ) , then
8~o(z) = 8/po(z)e-" o(z ) + /Po(z)( -Ao(z) - Bo(z) ::f:~ )e-tDo(z)
=(8cpo(z) -
Ao(z)cpo(z) - Bo(z)cpo(z»e-wo(z) = o.
Therefore, iJo(z) is a hypercomplex analytic function, as a result, every solution of the equation (15) has the following form (16)
where iJ(z) is a hypercomplex analytic function. w(z) is a continuous in the whole plane and hypercomplex analytic in (; and vanished at infinity. In view of following formula if 1 U E C (G ), then -
the
equ~tioD (14)
J1 G
t(Dt«)8u dG (zoE G t( () - t (Zo) ' ,
(17)
equivalent to following Fredholm integral equation
lfJo (%) where
1
1 u(z)dt(z) - 1 21ri r t( z) - t (zo) 1r
( ) -UZo
-!
11"
Jf
lG
91(Z) = _1
t,Dt(()rAo(()/po(() + Bo«()~ dG tee) _ t(z)
f
lfJo«()dt«) _
21ri Jr t«() - t(z)
.!. 11"
JJGf
, - 91
t,Dt(()go(() dG t«) - t(z) (.
() Z ,
(18)
(19)
Let us show that homogeneous equation (18) only have the trival solutuion. Let CPo be such asolution, then lfJo(z) satisfies the equation (15), therefore,
Liu: TWO-DIMENSIONAL SINGULAR INTEGRAL EQUATIONS
No.1
where
r1(
Fron~
49
, z ), r 2 « , z), are resolvent kernel (see [1]1. (19) and (20), we obtain
= 2~i Jr{}(l)(z,()cR()dt«) -
-~
J fG[{1(1)(z, ()go«) -
10(1)(Z,() 10(2)(z, (),
n(2)(z,()C)«)dt«)
0(2)(z, ()po«)]dG(
t(()~t( .. ) 1= o(lz - (1- 2/P)
= o(lz -
(J-2/ p )
From (14) and note that tpo is hypercomplex analytic in we obtain that
where 0(1)(Z, ()
•
O(2)(Z, ~)
O(l)(Z,
={ =
1
o
and vanished at infinity,
z E G,
t(,)-t(z)
{{1(2)(Z,
G,
Z
o
E
G;
z E G,
_
.0
z E G;
g*(z) = { ; : f fG[O(l)(Z,()g«() - 0(2)(z, ()g«()]dG"
z E G, z'E
G.
By the Plemelj-Soknotki's formula (see [2]), we have 1
1
f
C)«)dt«)
= 21l"i Jr t«() _ t(z) -
1
2~(z),z
E f.
As so, because of (13), we obtain following singular integral equation.
1 2(1 + A(z»~(z)
_ A(z) 211"i
f
1 1 f () + 2D(z)~(z) + 21ri J 0 1 (z, ()~«()dt«()
~«()dt«() t(z)
Jr t«) -
1 --2. f 1r~ Jr
r
+ D(z) f 21ri
iWdt(() t(z)
Jr t«) -
0(2)(z,()~«()dt«() = g*(z)
(22)
Take conjugate to (22) we obtain 1
--
1--
1
f
-
2(1 + A(z»~(z) + 2D(z)~(z) - 21l"i Jr 0(1)(z, ()~«()dt«()
50
ACTA MATHEMATICA SCIENTIA
A(z) [iR)d@ D(z) +211"i Jr t(() - t(Z) - 211"i
11
+-2. 11"1,
r
Vol. 16
f ~«()dt«() Jr «o - t(z)
-
0(2)(z, ')~«()dt(') = g*(z).
(23)
Let 'It = (~, i), we obtain the character of the equation system (22) and (23) as following
A(z)'1'(z)
B(z)
+ -11"1,.
Ir t (() - t () = r
'It«()dt«() z
*-
-(g (z),g"(z»,
(24)
where
A(z):=! ( l+A(z) 2 D(z)
B(z):=!
1- A(z)
__
( -D(z)
D(z) ) 1+-:1(z) , D(z) ) -l+A(z) ,
and det(A + B) = A(z), det(A(z) - B(z» = A(z). Therefore, the regular condition of the equation system (22) and (23) is that A(z), z E r, is invertible (see [3]) i.e. cptA(z)
1= 0, z
E
r.
(25)
If this condition satisfied, the index of the equation system (22) and (23) is IC
:=
!ind(A(z»r := ![arg(cPtA(z»]r, 11"
11"
therefore, we have proved following theorems Theorem 2 If cptA(z) -# 0, and IcptA(z)1 -# IcptD(z)l, then the adjoint problem (13) and (14) is Fredholm (Noether) problem, i.e. the homogeneous problem (13) and (14) has finite linear independent solutions and the solvable conditions of the inhomogeneous problem (13) and (14) are finite. Theorem 3 If cptA(z) #. 0, and IcptA(z)1 1= IcptD(z)l, then the equation (Sr is a Fredholm equation. Acknowledgment I thank my advisor Prof. Zhao Zhen for his constant support and help over these years. References 1 Gilbert R P, Hile G. Generalized Hypercomplex F\mction Theory. Transactions of the Americal. Mathmatical Society,1974,195. 2 Gilbert R P, Buchanan J L. First Order Elliptic System, A function theoretic approach.New York:Academic Press,1983. 3 Liu Chungen. On Riemann Boundary Problem System of. the Hypercomplex Function.Acta Scientiarum Naturalium Universitatis Nan Kaiensis, 1993,3. 4 SiXWl Huang.Ke Xue Tong Bao (Chinese),1986,'T.