In a regular domain

*G*, we consider the problem

$(A):\{\begin{array}{c}L[w]:=Dw+Aw+B\overline{w}=F\phantom{\rule{1em}{0ex}}\text{in}G,\hfill \\ Re(\overline{\lambda}w)=\gamma \phantom{\rule{1em}{0ex}}\text{on}\partial G.\hfill \end{array}$

(3)

We refer to this problem as boundary value problem (

*A*). Where the unknown

$w(z)=\{{w}_{ij}(z)\}$ is an

$m\times s$ complex matrix-valued function,

$Q=\{{q}_{ij}(z)\}$ is a Hölder-continuous function which is a self-commuting matrix with

$m\times m$ and

${q}_{k,k-1}\ne 0$ for

$k=2,\dots ,m$.

$A=\{{a}_{ij}(z)\}$ and

$B=\{{b}_{ij}(z)\}$ are commuting with

*Q*, which is

$Q({z}_{1})A({z}_{2})=A({z}_{1})Q({z}_{1}),\phantom{\rule{2em}{0ex}}Q({z}_{1})B({z}_{2})=B({z}_{1})Q({z}_{1}).$

It is assumed, moreover, that *Q* is commuting with $\overline{Q}$ and $\lambda (z)\in {C}^{1}(\mathrm{\Gamma})$ is commuting with *Q*, where $\mathrm{\Gamma}=\partial G$, $\lambda \overline{\lambda}=I$. In respect of the data of problem (*A*), we also assume that *A*, *B* and $F\in {L}^{p,2}(\mathbb{C})$ and $\gamma \in {C}_{\alpha}(\mathrm{\Gamma})$. If $F\equiv 0$, $\gamma \equiv 0$, we have homogeneous problem (
).

We refer to the adjoint, homogeneous problem (

*A*) as (

); it is given by

where

*ϕ* is a generating solution for the generalized Beltrami system ([[

11], p.109]),

${B}^{\ast}={\varphi}_{z}^{-1}\overline{{\varphi}_{z}}\overline{B}$,

$\frac{d\varphi}{ds}:=\frac{\partial \varphi}{\partial z}\frac{dz}{ds}+\frac{\partial \varphi}{\partial \overline{z}}\frac{d\overline{z}}{ds}$ and

*ds* is the arc length differential. From the Green identity for

*Q*-holomorphic functions (see [[

11], p.113]), we have

$Re[\frac{1}{2i}{\int}_{\mathrm{\Gamma}}\phantom{\rule{0.2em}{0ex}}d\varphi {w}^{\mathrm{\prime}}w-{\iint}_{G}{\varphi}_{z}({w}^{\mathrm{\prime}}L[w]-{L}^{\mathrm{\prime}}\left[{w}^{\mathrm{\prime}}\right]w)\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy]=0,$

(5)

where

${w}^{\mathrm{\prime}}$ is commuting with

*Q*. For

$L[w]=F$ and

${L}^{\mathrm{\prime}}[{w}^{\mathrm{\prime}}]=0$, this becomes

$\frac{1}{2i}{\int}_{\mathrm{\Gamma}}\phantom{\rule{0.2em}{0ex}}d\varphi (z){w}^{\mathrm{\prime}}(z)\lambda (z)\gamma (z)-Re({\iint}_{G}{\varphi}_{z}(z){w}^{\mathrm{\prime}}(z)F(z)\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy)=0.$

(6)

Since

${w}^{\mathrm{\prime}}$ satisfies the boundary condition

$Re\left(\frac{d\varphi}{ds}\lambda {w}^{\mathrm{\prime}}\right)=0,$

(7)

we have

${w}^{\mathrm{\prime}}=i{\lambda}^{-1}{\left(\frac{d\varphi}{ds}\right)}^{-1}\varkappa ,$

(8)

where *ϰ* is a real matrix commuting with *Q*.

The solutions to problem (

) may be represented by means of fundamental kernels in terms of a real, matrix density

*ϰ* as

$\begin{array}{rcl}{w}^{\mathrm{\prime}}(z)& =& {P}^{-1}{\int}_{\mathrm{\Gamma}}(d\varphi (\zeta ){\mathrm{\Omega}}^{(1)}(z,\zeta ){w}^{\mathrm{\prime}}(\zeta )-d\overline{\varphi (\zeta )}{\mathrm{\Omega}}^{(2)}(z,\zeta )\overline{{w}^{\mathrm{\prime}}(\zeta )})\\ =& i{P}^{-1}{\int}_{\mathrm{\Gamma}}({\mathrm{\Omega}}^{(1)}(z,\zeta ){\lambda}^{-1}(\zeta )+\overline{{\mathrm{\Omega}}^{(2)}(z,\zeta ){\lambda}^{-1}(\zeta )})\varkappa (\zeta )\phantom{\rule{0.2em}{0ex}}ds,\end{array}$

(9)

see ([[

14], p.543]). In (9),

*P* is a constant matrix defined by

$P(z)={\int}_{|z|=1}{(zI+\overline{z}Q)}^{-1}(I\phantom{\rule{0.2em}{0ex}}dz+Q\phantom{\rule{0.2em}{0ex}}d\overline{z})$

called

*P*-value for the generalized Beltrami system [

11]. Since

*ϰ* is a real matrix commuting with

*Q*, inserting the expression (9) into the boundary condition (7), we have

${\int}_{\mathrm{\Gamma}}{K}_{1}(\zeta ,z)\varkappa (\zeta )\phantom{\rule{0.2em}{0ex}}d{s}_{\zeta}=0,\phantom{\rule{1em}{0ex}}z,\zeta =\zeta (s)\in \mathrm{\Gamma},$

(10)

where

${K}_{1}(\zeta ,z)=-Re[i{P}^{-1}\lambda (z)\frac{d\varphi (z)}{ds}({\mathrm{\Omega}}^{(1)}(z,\zeta ){\lambda}^{-1}(z)+\overline{{\mathrm{\Omega}}^{(2)}(z,\zeta ){\lambda}^{-1}(z)})].$

The integral in (10) is to be taken in the Cauchy principal value sense. If we denote this equation in an operator form by

$\underset{\sim}{K}\varkappa =0$ and its adjoint by

${\underset{\sim}{K}}^{\mathrm{\prime}}f=0$, then it may be easily demonstrated that the index of (10) is

$\kappa =k-{k}^{\mathrm{\prime}}=0$. Here

*k* and

${k}^{\mathrm{\prime}}$ are dimensions of null spaces of

$\underset{\sim}{K}$ and

${\underset{\sim}{K}}^{\mathrm{\prime}}$ respectively. If

$\{{\varkappa}_{1},\dots ,{\varkappa}_{k}\}$ is a complete system of solutions of (10), putting each of this into (9), we obtain the solutions of problem (

). However, it is possible that some of these solutions may turn out to be trivial solutions, which occurs when

${(\lambda \frac{d\varphi}{ds})}^{-1}\varkappa $ takes on the boundary values of a

*Q*-holomorphic function

${\psi}_{j}$ on each component of boundary contours

${\mathrm{\Gamma}}_{j}$ in

${C}^{1,\alpha}(\mathbb{C})$ which is, moreover,

*Q*-holomorphic in the domain

${G}_{j}$ bounded by the closed contour

${\mathrm{\Gamma}}_{j}$. Let

$\{{\varkappa}_{1},\dots ,{\varkappa}_{{\ell}^{\mathrm{\prime}}}\}$ be solutions of equation (

10) to which linearly independent solutions (see [

15])

${w}_{1}^{\mathrm{\prime}},\dots ,{w}_{{\ell}^{\mathrm{\prime}}}^{\mathrm{\prime}}$ of problem (

) correspond, then the remaining solutions

$\{{\varkappa}_{{\ell}^{\mathrm{\prime}}+1},\dots ,{\varkappa}_{k}\}$ satisfy the boundary condition of the form

$\varkappa (z)=i\lambda (z)\frac{d\varphi}{ds}{\mathrm{\Phi}}^{-}(z)\phantom{\rule{1em}{0ex}}\text{on}\mathrm{\Gamma}.$

(11)

Here

${\mathrm{\Phi}}^{-}$ are meant to be

*Q*-holomorphic functions outside of

${G}^{-}:=G+\mathrm{\Gamma}$ and

$\mathrm{\Phi}(\mathrm{\infty})=0$. Hence the

*Q*-holomorphic functions satisfy the homogeneous boundary conditions

In a complex case, Vekua refers to problems of this type as being concomitant to (
) and denotes them by (
). Let ${\ell}_{\ast}$ be a number of linearly independent solutions of this problem. Obviously, ${\ell}^{\mathrm{\prime}}+{\ell}_{\ast}^{\mathrm{\prime}}=k$.

Let us now return to the discussion of problem (

*A*), where we assume that

$\varkappa =0$ in what follows. The solutions of this problem may be expressed in terms of the generalized Cauchy kernel as follows:

$w(z)={w}_{1}(z)+{w}_{2}(z)=\underset{\sim}{C}[\lambda \gamma ](z)+\underset{\sim}{C}[i\lambda \mu ](z),$

where

$\underset{\sim}{C}[\mathrm{\Phi}]={P}^{-1}{\int}_{\mathrm{\Gamma}}\phantom{\rule{0.2em}{0ex}}d\varphi (\zeta ){\mathrm{\Omega}}^{(1)}(z,\zeta )\mathrm{\Phi}(\zeta )-d\overline{\varphi (\zeta )}{\mathrm{\Omega}}^{(2)}(z,\zeta )\overline{\mathrm{\Phi}(\zeta )}$

(see [[

14], p.543]). From the Plemelj formulas, it is seen that the density

*μ* must satisfy the integral equation

${\gamma}_{0}={\int}_{\mathrm{\Gamma}}{K}_{1}(\zeta ,z)\mu (z)\phantom{\rule{0.2em}{0ex}}d{s}_{z},$

(13)

where

${\gamma}_{0}(\zeta )=\gamma (\zeta )-Re[\overline{\lambda (\zeta )}{w}_{1}^{+}(\zeta )]=-Re[\overline{\lambda (\zeta )}{w}_{1}^{-}(\zeta )].$

(14)

Problem (

) concomitant to problem (

) has the boundary condition

$Re[{\lambda}^{-1}(z){\mathrm{\Phi}}^{-}(z)]=0$ on Γ, where Φ is

*Q*-holomorphic outside

$G+\mathrm{\Gamma}$ and

$\mathrm{\Phi}(\mathrm{\infty})=0$. Denoting the numbers of linearly independent solutions of (

) and (

) by

*ℓ* and

${\ell}_{\ast}$ respectively, we have

$k=\ell +{\ell}_{\ast}$. In order that (13) is solvable, it is necessary and sufficient that the nonhomogeneous data

${\gamma}_{0}$ satisfy the auxiliary conditions

${\int}_{\mathrm{\Gamma}}{\varkappa}_{j}(\zeta ){\gamma}_{0}(\zeta )\phantom{\rule{0.2em}{0ex}}d{s}_{\zeta}=0\phantom{\rule{1em}{0ex}}(j=1,\dots ,k),$

(15)

where

${\varkappa}_{j}$ are solutions to integral equation (

10). These solutions may be broken up into two groups

$\{{\varkappa}_{1},\dots ,{\varkappa}_{{\ell}^{\mathrm{\prime}}}\}$ and

$\{{\varkappa}_{{\ell}^{\mathrm{\prime}}+1},\dots ,{\varkappa}_{k}\}$ such that

${\varkappa}_{j}=i\lambda (z)\frac{d\varphi}{ds}{w}_{j}^{\mathrm{\prime}}(z)$ for

$j=1,\dots ,{\ell}^{\mathrm{\prime}}$ and

${\varkappa}_{j}=i\lambda (z)\frac{d\varphi}{ds}{\mathrm{\Phi}}_{j}$ for

$j={\ell}^{\mathrm{\prime}}+1,\dots ,k$, where

$z\in \mathrm{\Gamma}$. Here

${w}_{j}^{\mathrm{\prime}}$ and

${\mathrm{\Phi}}_{j}$ are solutions of problems (

) and (

) respectively. The condition (15) for

${\gamma}_{0}$ given by (14) becomes

$-{\int}_{\mathrm{\Gamma}}{\varkappa}_{j}(\zeta ){\gamma}_{0}(\zeta )\phantom{\rule{0.2em}{0ex}}d{s}_{\zeta}=-i{\int}_{\mathrm{\Gamma}}\phantom{\rule{0.2em}{0ex}}d\varphi (\zeta )\lambda (\zeta ){w}_{j}^{\mathrm{\prime}}(\zeta )\gamma (\zeta )+Re[i{\int}_{\mathrm{\Gamma}}\phantom{\rule{0.2em}{0ex}}d\varphi (\zeta ){w}_{j}^{\mathrm{\prime}}(\zeta ){w}_{1}^{+}(\zeta )]$

for

$j=1,\dots ,{\ell}^{\mathrm{\prime}}$, whereas for

$j={\ell}^{\mathrm{\prime}}+1,\dots ,k$, we have

${\int}_{\mathrm{\Gamma}}{\varkappa}_{j}(\zeta ){\gamma}_{0}(\zeta )\phantom{\rule{0.2em}{0ex}}ds=Re[i{\int}_{\mathrm{\Gamma}}\phantom{\rule{0.2em}{0ex}}d\varphi (\zeta ){\mathrm{\Phi}}_{j}^{-}(\zeta ){w}_{1}^{-}(\zeta )]=0.$

Consequently, the conditions (15) are seen to hold if (6) (with $F=0$) holds. From the above discussion, one obtains a Fredholm-type theorem for problem (*A*).

**Theorem 1** *Non*-*homogeneous boundary problem* (*A*) *is solvable if and only if the condition* (6) *is satisfied*, ${w}^{\mathrm{\prime}}$ *being an arbitrary solution of adjoint homogeneous boundary problem* (
).

This theorem immediately implies the following.

**Theorem 2** *Non*-*homogeneous boundary problem* (*A*) *is solvable for an arbitrary right*-*hand side if and only if adjoint homogeneous problem* (
) *has no solution*.