Boundary value problems associated with generalized Q-holomorphic functions
© Hızlıyel; licensee Springer. 2013
Received: 9 November 2012
Accepted: 31 January 2013
Published: 18 February 2013
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.
Keywordsgeneralized Beltrami systems Q-holomorphic functions Riemann-Hilbert problem
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  and Bers . 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 .
where the unknown is an complex matrix, is a self-commuting complex matrix with dimension and for . and are commuting with Q. Solutions of such an equation were called generalized Q-holomorphic functions.
Solvability of Riemann-Hilbert problems
It is assumed, moreover, that Q is commuting with and is commuting with Q, where , . In respect of the data of problem (A), we also assume that A, B and and . If , , we have homogeneous problem ( ).
where ϰ is a real matrix commuting with Q.
In a complex case, Vekua refers to problems of this type as being concomitant to ( ) and denotes them by ( ). Let be a number of linearly independent solutions of this problem. Obviously, .
Consequently, the conditions (15) are seen to hold if (6) (with ) 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, 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.
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