Solvability of the conjugation boundary value problem for the Vekua nonlinear equation on a Riemann surface (Q2477456)

The boundary value problem of the linear conjugation \(w^+(t)-G(t)w^-(t)=g(t)\), \(t\in L\), for the Vekua equation \(\overline{\partial}w =wA(z,w)+\overline{w}B(z,w)+F(z,w)\), \(w\in D^\pm\), is considered. Here \(D\) is a compact Riemann surface of genus \(\rho\), \(L\) is a smooth closed contour on D dividing it into two parts \(D^\pm\), \(w^\pm\) are the boundary values of \(w^\pm=w| _{D^\pm}\). Earlier some partial cases of the problem (\(B\equiv0\), \(A\) and \(F\) don't depend on \(w\)) were investigated by Yu.~Rodin and I.~Bikchentaev. The case of analytic functions was considered in \textit{V. N. Monakhov} and \textit{E. V. Semenko} [Boundary value problems and pseudodifferential operators on Riemannian surfaces. Moskva: Fizmatlit. (2003; Zbl 1060.35002)]. The authors shows that under some natural conditions on the initial data the problem has a solution. The main tool of the proof is the Schauder fixed point principle.