On nonsummable solutions to an integral equation with logarithmic kernel (Q2713984)

The author considers the integral equation NEWLINE\[NEWLINE A(t)\mu(t)\int_{\breve{at}}\varphi(\tau) d\tau +\frac{B(t)}{\pi i}\int_{\Gamma} L_b(\tau,t)\varphi(\tau) d\tau =f(t), \tag{1} NEWLINE\]NEWLINE where \(A,B\) are Hölder functions, \(L_b(\tau,z)=\int_{\breve{\tau b}} \frac{\mu(\xi)}{\xi-z} d\xi\) (\(\mu\) is a Hölder function), \(\Gamma\) is a finite smooth curve joining the points \(a,b\in\mathbb C\), and the symbols \(\breve{a t}\) and \(\breve{t b}\) stand for the arcs of \(\Gamma\) joining the points \(a\) and \(t\) or \(t\) and \(b\). A solution \(\varphi\) to equation (1) is sought in the class \(\varphi(t)=\psi(t)/(t-a)^{1-\varepsilon}(b-t)^{2-\varepsilon}\), with \(\psi\) a Hölder function on \(\Gamma\) and \(\varepsilon>0\). Under some additional smoothness assumptions on the data of the problem it is demonstrated that a solution to (1) exists, and an exact representation formula for the solution is presented. The problem is reduced to a system of singular integral equations with Cauchy kernels.