Effective estimates for the number of solutions of a class of singular integral equations (Q1873518)

The author considers the problem of estimating partial indices \(\varkappa_1\), \(\varkappa_2\) of a \(2\times2\) matrix function \[ G(t)=\| k_jb_j\| _{2\times2}-E=G_-(t)\text{ diag}\{t^{\varkappa_1}, t^{\varkappa_2}\}\;G_+(t)\, ,\qquad t\in\Gamma\, ,\tag{1} \] where \(\Gamma\) is a simple smooth closed contour on the complex plane. Equation (1) is linked to the equivalent (in the sense of number of solutions) with the following singular integral equation ``with two kernels'' \[ K\varphi=a_0\varphi+a_1S[k_1\varphi]+a_2S[k_2\varphi]=f,\tag{2} \] where \(S=S_\Gamma\) is the Cauchy singular integral operator \[ S\varphi(t):=(\pi i)^{-1}\int\limits_\Gamma(\tau-t)^{-1} \varphi(\tau)\,d\tau \] and \[ b_j:=2a_j\Delta_1\, ,\qquad \Delta_1:=a_0+k_1s_1+k_2a_2\, , \quad j=1,2\, . \] The author gives estimates for the number of solutions of equation (2). In conclusion an example of rational matrix-function on the circle \(| z| =3/2\) is considered.