Explicit solutions of Cauchy singular integral equations with weighted Carleman shift (Q986576)

The authors consider the singular integral equation with shift in the Lebesgue space \(L_p(\Gamma),\) \(p \in (1,\infty)\) \[ f(t)\varphi(t) + g(t)(S_\Gamma \varphi)(t)+ g(t)v(t)(S_\Gamma \varphi)(\alpha(t)) = h(t), \] \[ (S_\Gamma f)(t) = \frac{1}{\pi i} \int\limits_{\Gamma} \frac{f(\tau)}{\tau-t}d\tau, \quad t \in \Gamma, \tag{1} \] where \(\Gamma\) is a simple closed smooth curve, \(\alpha(t)\) is a Carleman shift function, \(v(t) v(\alpha(t)) = 1\). Existence and uniqueness of solutions of the equation (1) are proved.