Existence of solutions for a class of quasi-linear singular integro-differential equations in a Sobolev space (Q730773)

The authors consider the quasi-linear singular integro-differential equation with Cauchy kernel \[ A(s,u(s))u'(s)-B(s,u(s))\frac{1}{\pi i}\int_{\Gamma}\frac{u'(\tau)}{\tau-s}\,d\tau=g(s,u(s)), \] subjected to the initial condition \[ u(r)=0 \] (for \( r\) arbitrary but fixed in \( \Gamma\)), in the Sobolev space \(W^1_p(\Gamma)\) and, by using Schauder's fixed point theorem, they prove the existence of at least one solution.