Turk J Math
34 (2010) , 167 – 180.
©c TÜBİTAK
doi:10.3906/mat-0804-22
The Riemann Hilbert problem for generalized Q-holomorphic
functions
Sezayi Hızlıyel and Mehmet Çağlıyan
Abstract
In this work, the classical Riemann Hilbert boundary value problem is extended to generalized Q-
holomorphic functions.
Key Words: Generalized Beltrami systems, Q-Holomorphic functions, Riemann Hilbert problem.
1. Introduction
In [6] A. Douglis developed an analogue of analytic functions theory for more general elliptic systems in
the plane of the form
wx + iwy + aEwx + bEwy = 0, (1)
where E is an m×m constant matrix, w is an m× 1 vector, and a and b are complex valued functions of x
and y . Subsequently in [5] B. Bojarskĭı extended the function theory of Douglis to a system which he wrote in
the form
wz = qwz. (2)
He assumed that the variable m×m matrix q is “lower diagonal with all eigenvalues of q having magnitude
less than 1. The systems (1) and (2) are natural ones to consider because they arise from the reduction of
general elliptic systems of first order in the plane to a standard canonical form.
Douglis and Bojarskĭı theory has been used to study the elliptic systems of more general form:
wz − qwz = aw + bw.
Solutions of this equation were called generalized (or pseudo) hyperanalytic functions. Works in this direction
appear in [7, 8, 10, 11]. These results extend the generalized (or “pseudo”) analytic function theory of Bers
[4] and Vekua [17]. Also, the classical boundary value problems for analytic functions were extended to the
generalized hyperanalytic functions. A good survey of the methods encountered in the hyperanalytic case may
be found in [3, 9], see also [1, 2].
AMS Mathematics Subject Classification: 30G20, 30G35, 35J55.
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HIZLIYEL, ÇAĞLIYAN
In [13], Hile noticed that what appears to be the essential property of the elliptic systems in the plane
for which one can obtain a useful extension of analytic function theory is the self commuting property of the
variable matrix Q , which means
Q (z1)Q (z2) = Q (z2)Q (z1)
for any two points z1, z2 in the domain G0 of Q . Further, such a Q matrix can not be brought into the
quasi-diagonal form of Bojarskĭı by a similarity transformation. So Hile [13] attempts to extend the results of
Douglis and Bojarskĭı to a wider class of systems in the same form as (2). If Q (z) is self-commuting in G0
and if Q (z) has no eigenvalues of magnitude 1 for each z in G0 , then Hile called the system (2) generalized
Beltrami system and the solutions of such a system are called Q-holomorphic functions. Later in [14, 15] using
Vekua and Bers techniques a function theory is given for the equation
wz −Qwz = Aw + Bw, (3)
where the unknown w(z) = {wij(z)} is an m×s complex matrix, Q(z) = {qij(z)} is a self commuting complex
matrix with m × m , and qk,k−1 = 0 for k = 2, . . .m . A = {aij(z)} and B = {bij(z)} are m × m-complex
matrices commuting with Q . Solutions of such equation were called generalized Q-holomorphic functions.
In this work, we consider the Riemann Hilbert boundary value problem for the equation (3) with the
boundary condition
Re(γw) = ϕ on ∂G,
where the coefficients A and B are Hölder continuous in a bounded simply connected region G with piecewise
Hölder continuous boundary. γ is commuting with Q . A and B are continuous in G ∪ ∂G . Moreover, γ
has one Hölder continuous derivative, ϕ is real Hölder continuous function on ∂G . Also we assume that Q
commute with Q .
2. Fundamental operators
To investigate Q -holomorphic functions, Hile introduced the notion of generating solution for generalized
Beltrami operator
∂ ∂
D := −Q . (4)
∂z ∂z
This generating solution can be written as φ (z) := φ0 (z) I +N (z) and satisfies the equation Dφ = 0, where
N is the nilpotent part of φ and φ0 is the main diagonal term of φ satisfying the Beltrami equation
∂φ0 − ∂φ0λ = 0,
∂z ∂z
where |λ(z)| = 1.
Hile also gave the following representation formula called the generalized Cauchy-Pompieu representation
for the m× s complex matrix-valued functions.
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HIZLIYEL, ÇAĞLIYAN
( )
Theorem 1 Let G be a regular subdomain of G0 , with Γ = ∂G and w be an m× s-matrix in C1 (G)∩C G
with bounded first derivatives in G∫. Then for z in G
w (z) = P−1 (φ (ζ)− φ (z))−1 dφ (ζ)w (ζ) (5)
Γ ∫∫ [ ]
−2iP−1 φζ ( −1ζ) (φ (ζ)− φ (z)) wζ (ζ) −Q (ζ)wζ (ζ) dξdη.
G
In (5) P is constant matrix defined by∫
P (z) = ( −1zI + zQ) (Idz +Qdz) . (6)
|z|=1
It is called P -value for (4) [13].
Using Beltrami homeomorphism ρ(z) = φ0(z), we may(write[ ] )∂ − ∂ ∂ ∂Q = ρz(λQ− I) − Q̂ ,
∂z ∂z ∂ρ ∂ρ
[ ]
− −1where Q̂ = ρz(λQ I) [ρz(λI −Q)] is self-commuting matrix whose the main diagonal terms are zero (see
[14], pp. 431). Note that for the equation in normal form the generating solution is φ(z) = zI +N(z) and the
complex Pompieu formula is
∫ ∫∫
w (ζ)
w (z) = P−1
w (ζ) −1 ζ
ζ − dζ − 2iP − dξdη, (7)z ζ z
∂G G
where N(z) is m×m-type nilpotent matrix (see [16], p∫p. 581). The operators
w (ζ)
Φ̃ −1∂Gw(z) = P dζ
ζ − z
∂G ∫∫
wζ(ζ)
T̃Gw(z) = −2iP−1 − dξdηζ z
G
and ∫∫
w (ζ)
Π̃Gw(z) = −2 −1 ζiP (ζ − dξdηz)2
G
have similar properties as Φ, T and Π-operators of Vekua’s theory and the following theorems concerning
Φ̃∂G , T̃G and Π̃G can be proved as in the book of Vekua [17].
Theorem 2 If G ∈ Cm m mα , then Φ̃∂G : Cα (∂G) → Cα (G) .
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HIZLIYEL, ÇAĞLIYAN
Theorem 3 If f ∈ L1(G) , then ∂zT̃Gf = f . If p > 2 , then T̃G : Lp(G) → C p−2 (C) . If G ∈ C,w ∈ C(G) ,
p
and ∂zw ∈ Lp(G), p > 2 , then
w(z) = Φ̃∂Gw(z) + T̃G(∂ζw)(z).
Theorem 4 If G ∈ Cm+1α , then T̃G : Cm(G) → Cm+1α α (G) and Π̃G : Cmα (G) → Cmα (G) . Moreover ∂zT̃G = Π̃G .
The operator Π̃G can be extended to a bounded linear operator on Lp(C), p > 1 , with ∂zT̃C = Π̃C .
As in the complex case (see [12], pp. 259), using the complex Pompieu formula, for any complex matrix-
valued function w that is C1(G) and Hölder continuous in G we have the representation
w (z) = Ω (z) + (Pf) (z) , wz(z) = f(z),
where
∫ [ ]
Ω (z) = −2πiP−1 dnGI (ζ, z)− idGII (ζ, z) Rew (ζ)
∫∂G
−2πP−1 d GIIn (ζ, z) Imw (ζ)
∂∫G∫ [ ]
(Pf) (z) = −2πP−1 GI (ζ, z) +GIIζ ζ (ζ, z) f (ζ) dζdζ
∫G∫ [ ]
−2πP−1 GI (ζ, z) +GII (ζ, z) f (ζ)dζdζ. (8)
ζ ζ
G
GI and GII are the first and second Green’s functions for G and dn denotes the differential in normal direction.
If θ is a conformal mapping of G onto the unit disk C0 , then the Green functions of first and second kinds
may be expressed as
∣∣ ∣∣
GI
−1 θ (ζ) − θ (z)
(ζ, z) : = log
2π ∣∣∣ ∣∣∣1− θ (ζ)θ (z) ∣( )∣
GII
−1
(ζ, z) : = log ∣∣( ∣θ (ζ)− θ (z)) 1− θ (ζ)θ (z) ∣ .
2π
Thus Pf has the representation
∫∫ ′ ∫∫
−1 θ (ζ) f (ζ)
′
−1 θ (ζ) f (ζ)(Pf) (z) = P − dζdζ + P θ (z) dζdζ (9)θ (ζ) θ (z) 1− θ (ζ)θ (z)
G G
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HIZLIYEL, ÇAĞLIYAN
and ∂∂z (Pf)(z) may be expressed as
∂ (Pf) (z) ( )
= Π̃f (z)
∂z ∫∫ ⎨⎪⎧ ⎫′ ⎪
−1 ′ ⎪⎩ θ (ζ) f (ζ) [ θ
′ (ζ) f (ζ)
= P θ (z)
− 2
+ ] ⎬
[ ( ) ( )] 2θ ζ θ z 1− θ (ζ)θ (z) ⎭⎪dζdζ.G
The operator P may still put in a more convenient form. If we introduce the inverse mapping z = ρ(t) := θ−1(t),
we have ( ′) ( )
(Pf) ( ′z) = T̃C0f (ρ) ρ (θ (z))− θ (z) T̃C\C0f1 (ρ1) ρ1 (θ (z))
where C0 is the unit disk and ( )
1
ρ1(z) : = ρ , ( z ∈ C \ C0)
1 (z )
f1 (ρ1(z)) : = f ρ1 (z) .
z
By writing Pf in above form we obtain certain imbedding properties of T̃ and Π̃
( ) ( )
Cα Pf, G ≤ M1 (α,G)Cα f, G (10)
( ) ( )
Cα Πf, G ≤ M1 (α,G)Cα f, G . (11)
3. The Riemann-Hilbert problem
We consider the problem
Dw = Aw +Bw in G, Re (γw) = ϕ on ∂G. (12)
where the coefficients A and B are Hölder continuous in a bounded simply connected region G with piecewise
Hölder continuous boundary. γ = γ0I + N(z) is commuting with Q . A and B are continuous in G ∪ ∂G .
Moreover, γ has one Hölder continuous derivative and ϕ is real Hölder continuous function on ∂G . We assume
Q commute with Q . It is natural for us define the index o∫f this problem as
1
κ := indγ := d arg γ0.2π
∂G
Case 1. κ = 0. In the case of index zero, we may reduce our problem by setting ω = γw to the case
Dω = Ãω + B̃ω in G,
Reω = ϕ on ∂G,
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HIZLIYEL, ÇAĞLIYAN
where
à = A + γ−1Dγ, B̃ = γBγ−1.
To see that such a transformation is valid it is necessary to demonstrate that the inverse γ−1 exits in G .
This follows directly from the fact that |γ0| = 0 on ∂G , and hence it is possible to continue harmonically the
component of γ := R exp (−iθ) into interior of G such that R0 := |γ0| nowhere vanishes. Our problem may
be written as the system
Dωk = Ãkkωk + B̃kkωk + fk in G
Reωk = ϕk on ∂G, k = 1, · · · , m,  = 1, · · · , s (13)
where
k∑−1( )∂ω
= 0 = jf1 , fk qkj + Ãkjωj + B̃kjωj , (2 ≤ k ≤ m, 1 ≤  ≤ s).
∂z
j=1
The problem (13), which may be solved successively, may be replaced by integral equation by using Green
functions GI(ζ, z), GII(ζ, z) of first and second kinds respectively. We obtain
( )
ωk = Ωk +P Ãkkωk + B̃kkωk + fk ,
where Ωk is an analytic function given by∫
− [ ]Ω 1 I IIk = −2πP dnG (ζ, z)− idG (ζ, z) ϕk + ck (14)
∂G
and ck is arbitrary constant whic∫h can be fixed by setting it equal to boundary norm,
c = 2πP−1k Im (ω IIk) dnG (ζ, z) , k = 1, · · · , m,  = 1, · · · , s.
∂G
Taking advantage of the fact that Q is nilpotent yields a concise representation for ω as
[
m∑−1 ( )k ( )]′
ω = Ω+P QΠ̃ Ãω + B̃ω +QΩ ,
k=0
where ∑m ∑s ′ ∂Ω ∂P
Ω = Ωkek, Ω = and Π̃ = .
∂z ∂z
k=1 =1
and ek denotes m × s constant matrix in which k − th row and  − th column terms are 1 and the others
terms are 0. Moreover, by introducing
( )− m∑−11 ( )k
R := P I −QΠ̃ = P QΠ̃ , (15)
k=0
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HIZLIYEL, ÇAĞLIYAN
we obtain the matrix integral equation ( ′) ( )
ω = Ω−R QΩ +R Ãω + B̃ω . (16)
Using imbedding properties (10) and (11) we have following imbedding property in Cα(G)
( ) Mm (α,G)− 1 ( )
Cα Rf, G ≤ M 31 (α,G) − Cα f, G , (17)M3 (α,G) 1
where ( )
M3 (α,G) := (m− 1)M2 (α,G)Cα Q,G , (18)
and the norm Cα(f, G) is ∑m ∑s ( )
Cα(f, G) = Cα fk, G .
l=1 =1
The operator R is then seen from (15), (17) and (18) to be compact in Cα , hence is a Fredholm integral
operator. To show that integral equation (16) has a unique solution we consider the homogenous version of
(12) i.e. Reω |∂G= 0, c = 0. This means, since ReΩ |∂G= 0, that Ω = 0 and the Fredholm integral equation
corresponding to homogenous Riemann Hilbert prob(lem is hom)ogenous integral equation
ω = R Ãω + B̃ω .
It is easily seen that this homogenous integral equation has only trivial solution. This discussion is then
summarized as
Theorem 5 For any given real, m×s matrix-valued function ϕ ∈ Cα(∂G) and given real, m×s-type constant
matrix c there exits a unique solution w of the integral equation (16) which satisfies
∫
Im (ω) dnGII (ζ, z) = c.
∂G
Case 2. κ < 0. We assume in present case the i∫ndex is a negative integer
1
κ := indγ := d arg γ0 = −n.2π ∂G
We introduce a transformation ∏n
v := ψ−1w, ψ = [φ (z)− φ (zτ )] ,
τ=1
where the points zτ lie in G . The boundary value problem becomes
Dv = Âv + B̂v in G, Re (ρv) = ϕ on ∂G,
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HIZLIYEL, ÇAĞLIYAN
where
 = A, B̂ = ψ−1Bψ and ρ = γψ.
This reduced Riemann Hilbert problem has index zero, and we have modified the problem to case previously
discussed.
The homogenous boundary value problem for v which is normalized such that
∫
ch = −2πiP−1 Imvh(ζ)d IInG (ζ, z)
∂G
has non-trivial solution vh . That exits a non-trivial solution to this problem can be seen by considering 1− th
row and − th column of vh
n
∂v ∏h1 z − z= τÂ11vh1 + B̂11 vh1
∂z z − z
τ=1 τ
Re (ρ11vh1) = 0 , Indρ11 = 0
which is known to have a solution non vanishing in G (see [12], 11.1).
Now, for fixed , 1 ≤  ≤ s , we consider the boundary value problem
(Dw) = Aw + Bw in G
Re γw = ϕ on ∂G, (19)
∑ ∑
where w = m w ei and ϕ m ii=1 i = i=1 ϕie are m × s matrix-valued functions. Hence the homogenous
solutions wh of (19) with index κ = −n has a representation of the form
wh = λ ψv

 h , ∑
where λ is a real constant matrix commuting with Q and v =
m i
h i=1 vhie is a solution of
Dv = Âv + B̂v, Re(ρv) = 0.
Note that if λ commutes with Q then λ can be written as
∑m
λ = Pkλk1,
k=1
∑ ∑
where P1 =
m i−1
I, Pk = ij1=1 j=1 (Cij)k e , (2 ≤ k ≤ m) and (Ck,k−l)μ are real or complex constants such that{
( ) = 1, μ = kCk1 μ 0, μ = k
q
( k,k−1Ck,k−1)2 = q21
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HIZLIYEL, ÇAĞLIYAN
(C⎧k,k−l)μ⎨ 1 l∑−1 ∣∣∣∣ ∣(Ck,k−s)2 (C⎩ k,k−s
) ∣μ ∣
: = (Ck−l,k−l−1) s=1 (
for μ = 3, . . . , l+ 1
C ∣
2 k−s,k−l−1)2 (Ck−s,k−l−1)μ
0 for μ > l+ 1
l+1
q
( ) = k,k−l − 1
∑ qμ1
Ck,k−l 2 (C( ) k,k−l
)
q C − − − μ21 k l,k l 1 2 qμ=3 21
l+1
a
= k,k−l − 1
∑ a
( μ1Ck,k−l)
a μ21 (Ck−l,k−l−1)2 aμ=3 21
(see [14], pp.442). Moreover, it is clear that λ1 and λ2 commute with Q then λ1 commutes with λ2 .
From this it is easily seen that each homogenous solutions of (19) having n +∑1 distinct zeros must be
vanish identically in G . The general solution to (12) may be written as w = ψ( sv 0 + =1 λvh), where v0 is a
particular solution of reduced equation and λ are real constant matrices commuting with Q .
If w1 , w

2 are distinct solutions of
Dw = Aw +Bw in G,
Reγw = 0 on ∂G (20)
with negative index, then any combination of them
w = λ1w1 + λ2w

2
with real constant matrices λ1 and λ2 commuting with Q is also a solution of (20). The general solution of
(20) contains 2n arbitrary real constant zτ = xτ + iyτ , (1 ≤ τ ≤ n) and an arbitrary real constant matrix
λ . It may therefore be conjectured that there are 2n+ 1 linearly independent solution of (20).
r solutions w1, · · · , wr of (20) are said to be linearly independent if the equation
∑r
λ jwj = 0, (λj commuting with Q)
j=1
implies that λj = 0.
Suppose that we already know (2n+ 1) linearly independent solutions w̃0, w̃1, · · · , w̃2n of (20). No pair
of these solutions can have the same zeros. To show that there are no more than (2n+ 1) solutions, with non
vanishing 1 − th row and  − th column terms , we show that each such solution can be written as a linear
combination of w̃j, (0 ≤ j ≤ 2n). To this end let
∑m ∑i
(0)
λ(0) ijμ := λμije
i=1 j=1
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HIZLIYEL, ÇAĞLIYAN
be non-trivial solution of the system
∑2n
λ(0)w̃μ μ(zτ ) = 0, 1 ≤ τ ≤ n.
μ=0
In component form this becomes
∑2n ∑i
(0)
λμijw̃μj(zτ ) = 0, (1 ≤ i ≤ m, 1 ≤  ≤ s, 1 ≤ τ ≤ n)
μ=0 j=1
For each μ we define the functions ∑2n
 := (μ)wμ λ

k w̃k (z) ,
k=1
where (μ)λk are real constant matrices commuting with Q that are uniquely determined as the solution of system
w2μ (z
1  1
τ ) = δμτe , w2μ−1 (zτ ) = iδμτ e .
In complex form this becomes
∑2n ∑t
(2μ)
λktj wkj (zτ ) = δ1tδμτ ,
k=1 j=1
∑2n ∑t
(2μ−1)
λktj wkj (zτ ) = iδ1tδμτ ,
k=1 j=1
1 ≤ t ≤ m, 1 ≤ μ, τ ≤ n . To show that these inhomogeneous equations have a unique solution it is sufficient
to demonstrate that the system ∑2n
χ wk k (zτ ) = 0
k=1
possesses only trivial solution, where χk commutes with Q . This follows directly the fact that if
∑2n
χkw̃

k (zτ )
k=1
is a solution of (20), that is, linearly independent of w0 , it must be trivial solution. From this we conclude,
since w̃k are linearly independent, that the χk are all zero. In this way we construct a system of functions w

μ ,
μ = 0, · · · , 2n satisfying the conditions
w0 (zτ ) = 0, w
 1  1
2μ (zτ ) = δμτe , w2μ−1 (zτ ) = iδμτ e ,
1 ≤ t ≤ m, 1 ≤ μ, τ ≤ n . It is seen that each solution of (20) may be represented as a linear com∑bination of
w̃μ . For instance if we set (λ
 2n 
2τ )k1 = Rewk(zτ ) and (λ2τ−1)k1 = Imwk(zτ ) then the function w − μ=1 λμwμ
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HIZLIYEL, ÇAĞLIYAN
is a solution of (20) and it vanishes at zτ , τ = 1, · · · , n . Where (· · · )k1 means the k − th row and 1 − th
column elements of (· · · ). Therefore there is a unique real constant matrix commuting with Q such that
∑2n
w(z) − λμwμ(z) = λ w0 0 (z) .
μ=1
We can, moreover, show that there are at most (2n+ 1) linearly independent solutions. To this end, we
introduce w  2μ,τ and w2μ−1,τ as linear combinations of w̃ :
∑2n
w : = (2μ,t)λ w̃2μ,t k k
∑k=12n
 : = (2μ−1,t)w 2μ−1,t λk w̃k
k=1
which moreover satisfy the conditions
w t  t2μ,t (zτ ) := δμτe , w2μ−1,t (zτ ) := iδμτ e .
These two conditions for determining (2μ,t) , (2μ−1,t)λk λk may be formulated in terms of their components as
∑2n ∑l
(2μ,t)
λklj w̃kj (zτ ) = δtlδμτ
k=1 j=1
∑2n ∑l
(2μ−1,t)
λklj w̃kj (zτ ) = iδtlδμτ
k=1 j=1
1 ≤ t, l ≤ m , 1 ≤ μ, τ ≤ n . If we fix w0,t := (w0)t (1 ≤ t ≤ m), then each solution of (20) may be written as
a linear combination of wμ,t(z).
Now we show that there are exactly (2n+ 1) linearly independent solutions of (20). Let w0(z) be non-
trivial solution of (20) that vanishes at each of given points zτ (1 ≤ τ ≤ n). If we have 2n additional solutions
that moreover satisfy the conditions
w2k(zτ ) = δ
1  1
kτe , w2k−1(zτ ) = iδkτe , 1 ≤ k, τ ≤ n,
then these solutions also have non-vanishing 1−th row and −th column terms and form a linearly independent
system with w0(z).
Let us define ∏n
fk := [φ (z) − φ (zτ )] .
τ=1
τ =k
Then if w is a solution of (20), v = f−1wk is a solution of the homogenous boundary value problem of index
−1, ( )
Dv = Av + f−1k Bf kv in G, Re γf

kv = 0 on ∂G. (21)
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HIZLIYEL, ÇAĞLIYAN
If va, v

b are two non-trivial solutions of (21) that satisfy the boundary conditions Rev

a = Imv

b = 0 on ∂G ,
then it is known (see [12], pp.272) that without loss of generality we may assume that Imva1 > 0, Revb1 > 0
on ∂G . Furthermore let v (z), vα β(z) be two solutions of (21) that satisfy v

α(zτ ) = v

β(zτ ) = 0 and the
inhomogeneous boundary conditions ( ) ( )
Re (γf kvα) = −Re (γfkva) on ∂G
Re γf vk β = −Re γf vk b on ∂G.
Then the two functions ω1 := fk(v

a + vα), ω2 := fk(v

b + v

β) are solutions of (20) that satisfy
ω1 (z

τ ) = ω2 (zτ ) = 0, τ = k, 1 ≤ τ ≤ n
ω1[(zk) = fk (zk) v
  
a (z]k) = 0, ω2 (zk) = fk (zk) vb (zk) = 0.
Consequently, one has Im (ω1)1 (zk) (ω2)1 (zk) = 0 and the linear equations
(1)
λ ω2n 1 (
(2)
z ) + λ  1k 2nω2 (zk) = e
(1) (2)
λ2n−1ω
  1
1 (zk) + λ2n−1ω2 (zk) = ie
may be solved for real constant matrices (1)λ2n ,
(2) (1) (2)
λ2n , λ2n−1 , λ2n−1 , commuting with Q , having non vanishing
main diagonal terms. In this way we may construct two solutions
w = λ(1)ω + λ(2) μ μ 1 μ ω2 , μ = 2n− 1, 2n
with the properties
w 1 2k (zτ ) = δkτe , w2k−1 (zτ ) = iδ e
1
kτ .
By doing this for each k , we obtain 2n+ 1 linearly independent solutions. Hence, the homogenous boundary
value problem (20) has exactly (2n + 1)m linearly independent solutions over R . This discussion is then
summarized as the next theorem.
Theorem 6 The homogenous boundary value problem
Dw = Aw +Bw in G, Re (γw) = 0 on ∂G
has exactly (2n + 1)ms linearly independent solutions with non-identically vanishing 1 − th row and  − th
column terms of w over R .
Case 3. κ > 0. We consider the boundary value problem
Dw = Aw +Bw in G, Re (γw) = ϕ on ∂G.
We assume in present case the index is a positive integ∫er
1
κ := Indγ = d argγ0 = n.2π ∂G
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HIZLIYEL, ÇAĞLIYAN
We introduce a transformation ∏κ −1
ω = ψw, ψ−1 := [φ (z) − φ (zτ )] ,
τ=1
where the points zτ lie in G . Then the new boundary val(ue probl)em becomes
Dω = Ãω + B̃ω in G, Re γψ−1ω = ϕ on ∂G,
where
à = A, B̃ = ψBψ−1 .
Each of non-trivial solutions of homogenous boundary value problem with integral index has no zeros on
the boundary ∂G . Since w1 and γ0 are perpendicular on ∂G , w1 has the same index as γ0 . So, with each
solution w of differential equation
Dw = Aw + Bw
which has no zeros on ∂G , i.e. w = 0 on ∂G , we can always associate a homogenous boundary value problem
of integral index. Hence the index ∫
1
n = dargw
2 1π
∂G
of a solution w of homogenous differential equation
Dw = Aw +Bw,
which has neither zeros nor poles on ∂G, is equal to the difference between the numbers of its poles and zeros
in G . Hence every such solution w has a representation of the form
∑s
w = λ ψwh
=1
and the poles and zeros of ψ coincide with those of w .
We clearly do not investigate the conditions under which there are continuous solution in G even when
the boundary vector family has positive index
References
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HIZLIYEL, ÇAĞLIYAN
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Sezayi HIZLIYEL, Mehmet ÇAĞLIYAN Received 24.04.2008
Uludağ University, Art and Science Faculty
Deparment of Mathematics
16059 Görükle-Bursa, TURKEY
e-mail : hizliyel@uludag.edu.tr
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