- Open Access
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.
- generalized 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.
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.
- Douglis A: A function theoretic approach to elliptic systems of equations in two variables. Commun. Pure Appl. Math. 1953, 6: 259-289. 10.1002/cpa.3160060205MathSciNetView ArticleGoogle Scholar
- Bojarskiĭ BV: Theory of generalized analytic vectors. Ann. Pol. Math. 1966, 17: 281-320. (in Russian)MathSciNetGoogle Scholar
- Gilbert RP Lecture Notes Math. 365. In Constructive Methods for Elliptic Equations. Springer, Berlin; 1974.Google Scholar
- Gilbert RP, Hile GN: Generalized hypercomplex function theory. Trans. Am. Math. Soc. 1974, 195: 1-29.MathSciNetView ArticleGoogle Scholar
- Gilbert RP, Hile GN: Hypercomplex function theory in the sense of L. Bers. Math. Nachr. 1976, 72: 187-200. 10.1002/mana.19760720117MathSciNetView ArticleGoogle Scholar
- Vekua IN: Generalized Analytic Functions. Pergamon, Oxford; 1962.Google Scholar
- Bers L: Theory of Pseudo-Analytic Functions. Inst. Math. Mech., New York University, New York; 1953.Google Scholar
- Begehr H, Gilbert RP: Transformations, Transmutations, and Kernel Functions. Longman, Harlow; 1992.Google Scholar
- Gilbert RP, Buchanan JL: First Order Elliptic Systems: A Function Theoretic Approach. Academic Press, Orlando; 1983.Google Scholar
- Begehr H, Gilbert RP: Boundary value problems associated with first order elliptic systems in the plane. Contemp. Math. 1982, 11: 13-48.View ArticleGoogle Scholar
- Hile GN: Function theory for generalized Beltrami systems. Compos. Math. 1982, 11: 101-125.Google Scholar
- Hızlıyel S, Çağlıyan M: Generalized Q -holomorphic functions. Complex Var. Theory Appl. 2004, 49: 427-447.MathSciNetGoogle Scholar
- Hızlıyel S, Çağlıyan M: Pseudo Q -holomorphic functions. Complex Var. Theory Appl. 2004, 49: 941-955.MathSciNetGoogle Scholar
- Hızlıyel S: The Hilbert problem for generalized Q -homomorphic functions. Z. Anal. Anwend. 2006, 25: 535-554.MathSciNetGoogle Scholar
- Hızlıyel S, Çağlıyan M: The Riemann Hilbert problem for generalized Q -homomorphic functions. Turk. J. Math. 2010, 34: 167-180.Google Scholar
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