This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Boundary value problems associated with generalized Q-holomorphic functions

Sezayi Hızlıyel

Author Affiliations

Department of Mathematics, Faculty of Art and Science, Uludağ University, Görükle, Bursa, Turkiye

Boundary Value Problems 2013, 2013:33  doi:10.1186/1687-2770-2013-33


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/33


Received:9 November 2012
Accepted:31 January 2013
Published:18 February 2013

© 2013 Hızlıyel; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this work, we discuss Riemann-Hilbert and its adjoint homogeneous problem associated with generalized Q-holomorphic functions and investigate the solvability of the Riemann-Hilbert problem.

Keywords:
generalized Beltrami systems; Q-holomorphic functions; Riemann-Hilbert problem

Introduction

Douglis [1] and Bojarskiĭ [2] developed an analog of analytic functions for elliptic systems in the plane of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M1">View MathML</a>

(1)

where w is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M2">View MathML</a> vector and q is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M3">View MathML</a> quasi-diagonal matrix. Also, Bojarskiĭ assumed that all eigenvalues of q are less than 1. Such systems are natural ones to consider because they arise from the reduction of general elliptic systems in the plane to a standard canonical form. Subsequently Douglis and Bojarkii’s theory has been used to study elliptic systems in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M4">View MathML</a>

and the solutions of such equations were called generalized (or pseudo) hyperanalytic functions. Work in this direction appears in [3-5]. These results extend the generalized (or ‘pseudo’) analytic function theory of Vekua [6] and Bers [7]. Also, classical boundary value problems for analytic functions were extended to generalized hyperanalytic functions. A good survey of the methods encountered in a hyperanalytic case may be found in [8,9], also see [10].

In [11], Hile noticed that what appears to be the essential property of 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M5">View MathML</a>

for any two points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M7">View MathML</a> in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M8">View MathML</a> of Q. Further, such a Q matrix cannot be brought into a quasi-diagonal form of Bojarskiĭ by a similarity transformation. So, Hile [11] attempted to extend the results of Douglis and Bojarskiĭ to a wider class of systems in the same form with equation (1). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M9">View MathML</a> is self-commuting in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M8">View MathML</a> and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M9">View MathML</a> has no eigenvalues of magnitude 1 for each z in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M8">View MathML</a>, then Hile called the system (1) the generalized Beltrami system and the solutions of such a system were called Q-holomorphic functions. Later in [12,13], using Vekua and Bers techniques, a function theory is given for the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M13">View MathML</a>

(2)

where the unknown <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M14">View MathML</a> is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M15">View MathML</a> complex matrix, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M16">View MathML</a> is a self-commuting complex matrix with dimension <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M18">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M19">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M21">View MathML</a> are commuting with Q. Solutions of such an equation were called generalizedQ-holomorphic functions.

In this work, as in a complex case, following Vekua (see [[6], pp.228-236]), we investigate the necessary and sufficient condition of solvability of the Riemann-Hilbert problem for equation (2).

Solvability of Riemann-Hilbert problems

In a regular domain G, we consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M22">View MathML</a>

(3)

We refer to this problem as boundary value problem (A). Where the unknown <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M23">View MathML</a> is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M15">View MathML</a> complex matrix-valued function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M25">View MathML</a> is a Hölder-continuous function which is a self-commuting matrix with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M18">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M19">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M30">View MathML</a> are commuting with Q, which is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M31">View MathML</a>

It is assumed, moreover, that Q is commuting with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M33">View MathML</a> is commuting with Q, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M35">View MathML</a>. In respect of the data of problem (A), we also assume that A, B and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M37">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M39">View MathML</a>, we have homogeneous problem ().

We refer to the adjoint, homogeneous problem (A) as (); it is given by

(4)

where ϕ is a generating solution for the generalized Beltrami system ([[11], p.109]), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M44">View MathML</a> and ds is the arc length differential. From the Green identity for Q-holomorphic functions (see [[11], p.113]), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M45">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M46">View MathML</a> is commuting with Q. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M48">View MathML</a>, this becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M49">View MathML</a>

(6)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M46">View MathML</a> satisfies the boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M51">View MathML</a>

(7)

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M52">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M54">View MathML</a>

(9)

see ([[14], p.543]). In (9), P is a constant matrix defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M55">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M56">View MathML</a>

(10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M57">View MathML</a>

The integral in (10) is to be taken in the Cauchy principal value sense. If we denote this equation in an operator form by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M58">View MathML</a> and its adjoint by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M59">View MathML</a>, then it may be easily demonstrated that the index of (10) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M60">View MathML</a>. Here k and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M61">View MathML</a> are dimensions of null spaces of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M63">View MathML</a> respectively. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M64">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M66">View MathML</a> takes on the boundary values of a Q-holomorphic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M67">View MathML</a> on each component of boundary contours <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M68">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M69">View MathML</a> which is, moreover, Q-holomorphic in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M70">View MathML</a> bounded by the closed contour <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M71">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M72">View MathML</a> be solutions of equation (10) to which linearly independent solutions (see [15]) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M73">View MathML</a> of problem () correspond, then the remaining solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M75">View MathML</a> satisfy the boundary condition of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M76">View MathML</a>

(11)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M77">View MathML</a> are meant to be Q-holomorphic functions outside of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M79">View MathML</a>. Hence the Q-holomorphic functions satisfy the homogeneous boundary conditions

(12)

In a complex case, Vekua refers to problems of this type as being concomitant to () and denotes them by (). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M83">View MathML</a> be a number of linearly independent solutions of this problem. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M84">View MathML</a>.

Let us now return to the discussion of problem (A), where we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M85">View MathML</a> in what follows. The solutions of this problem may be expressed in terms of the generalized Cauchy kernel as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M86">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M87">View MathML</a>

(see [[14], p.543]). From the Plemelj formulas, it is seen that the density μ must satisfy the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M88">View MathML</a>

(13)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M89">View MathML</a>

(14)

Problem () concomitant to problem () has the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M92">View MathML</a> on Γ, where Φ is Q-holomorphic outside <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M79">View MathML</a>. Denoting the numbers of linearly independent solutions of () and () by and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M83">View MathML</a> respectively, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M98">View MathML</a>. In order that (13) is solvable, it is necessary and sufficient that the nonhomogeneous data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M99">View MathML</a> satisfy the auxiliary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M100">View MathML</a>

(15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M101">View MathML</a> are solutions to integral equation (10). These solutions may be broken up into two groups <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M103">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M104">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M106">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M107">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M108">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M110">View MathML</a> are solutions of problems () and () respectively. The condition (15) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M113">View MathML</a> given by (14) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M114">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M105">View MathML</a>, whereas for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M116">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M117">View MathML</a>

Consequently, the conditions (15) are seen to hold if (6) (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M118">View MathML</a>) holds. From the above discussion, one obtains a Fredholm-type theorem for problem (A).

Theorem 1Non-homogeneous boundary problem (A) is solvable if and only if the condition (6) is satisfied, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/33/mathml/M46">View MathML</a>being an arbitrary solution of adjoint homogeneous boundary problem ().

This theorem immediately implies the following.

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

Competing interests

The author declares that they have no competing interests.

References

  1. Douglis, A: A function theoretic approach to elliptic systems of equations in two variables. Commun. Pure Appl. Math.. 6, 259–289 (1953). Publisher Full Text OpenURL

  2. Bojarskiĭ, BV: Theory of generalized analytic vectors. Ann. Pol. Math.. 17, 281–320 (in Russian) (1966)

  3. Gilbert, RP: Constructive Methods for Elliptic Equations, Springer, Berlin (1974)

  4. Gilbert, RP, Hile, GN: Generalized hypercomplex function theory. Trans. Am. Math. Soc.. 195, 1–29 (1974)

  5. Gilbert, RP, Hile, GN: Hypercomplex function theory in the sense of L. Bers. Math. Nachr.. 72, 187–200 (1976). Publisher Full Text OpenURL

  6. Vekua, IN: Generalized Analytic Functions, Pergamon, Oxford (1962)

  7. Bers, L: Theory of Pseudo-Analytic Functions, Inst. Math. Mech., New York University, New York (1953)

  8. Begehr, H, Gilbert, RP: Transformations, Transmutations, and Kernel Functions, Longman, Harlow (1992)

  9. Gilbert, RP, Buchanan, JL: First Order Elliptic Systems: A Function Theoretic Approach, Academic Press, Orlando (1983)

  10. Begehr, H, Gilbert, RP: Boundary value problems associated with first order elliptic systems in the plane. Contemp. Math.. 11, 13–48 (1982)

  11. Hile, GN: Function theory for generalized Beltrami systems. Compos. Math.. 11, 101–125 (1982)

  12. Hızlıyel, S, Çağlıyan, M: Generalized Q-holomorphic functions. Complex Var. Theory Appl.. 49, 427–447 (2004)

  13. Hızlıyel, S, Çağlıyan, M: Pseudo Q-holomorphic functions. Complex Var. Theory Appl.. 49, 941–955 (2004)

  14. Hızlıyel, S: The Hilbert problem for generalized Q-homomorphic functions. Z. Anal. Anwend.. 25, 535–554 (2006)

  15. Hızlıyel, S, Çağlıyan, M: The Riemann Hilbert problem for generalized Q-homomorphic functions. Turk. J. Math.. 34, 167–180 (2010)