LAUSR

ABSTRACT In this paper, we study Riemann–Hilbert problems with Carleman shift and conjugation on the Lyapunov curve for null solutions of some certain polynomial Beltrami equations. Firstly, by constructing and verifying an explicit weakly singular kernel via a shift transformation between Lyapunov curves, and using the Fredholm integral equations method based on Cauchy's formula associated with the Beltrami equations, new integral representations of these generalized analytic functions with order one are obtained. Next, we introduce the notion of the canonical matrix in the context of some Beltrami equations and develop its theory, especially the method of constructing the explicit canonical matrix for triangular matrix functions. Finally, by using the decomposition theorem, the conversion method as well as constructing the canonical matrix, we obtain explicit formulae of solutions and conditions of solvability for the problems mentioned above.