Generalized Riemann Hilbert boundary value problem. (Q2756005)

For \(L\) a simple closed rectifiable curve dividing the complex plane \(\mathbb C\) into domains \(D^+\) and \(D^-\), NEWLINE\[NEWLINE(S\phi)(t)={1\over\pi i}\int_L{\phi(\tau)-\phi(t)\over\tau-t}d\tau+\phi(t),\quad t\in L,NEWLINE\]NEWLINE \(P=(I+S)/2, Q=(I-S)/2\), the authors consider the operator equation \((A_1PA_2P\cdots A_nP+B_1QB_2Q\cdots B_nQ)\phi=f\). The functions \(A_1,A_2,\cdots,A_n,B_1,B_2,\cdots,B_n\) belong to a certain functional algebra. The authors give an explicit solution of the equation using the theory of the Riemann-Hilbert boundary value problem.