The solvability of some kinds of singular integral equations of convolution type with variable integral limits (Q6616531)

Let \(\widehat{H}(\overline{\mathbb{R}})\) denote the space of Hölder continuous functions on \(\overline{\mathbb{R}}=[-\infty,+\infty]\) and let \(H_1\) be the set of functions \(F^{-1}\varphi\) for \(\varphi\in L^2(\mathbb{R})\cap\widehat{H}(\overline{\mathbb{R}})\), where \(F^{-1}\varphi(\tau)=\frac{1}{\sqrt{2\pi}} \int_\mathbb{R}e^{-is\tau}\varphi(s)ds\). Let \(A,B,C,D\in\mathbb{C}\) and \(h,k,n,h_1,h_2\in H_1\). The paper is devoted to the construction of the solution of the following singular integral equations: \N\[\NA\psi(\tau)+\frac{B}{\sqrt{2\pi}}\int_0^\tau \psi(t)h(\tau-t)dt +\frac{C}{\sqrt{2\pi}}\int_{\tau}^0 \psi(t)k(\tau-t)dt +\frac{D}{\pi i}\int_\mathbb{R}\frac{\psi(t)}{t-\tau}dt=n(\tau),\N\]\Nwith \(\tau\in\mathbb{R}\) and \N\[\NA\psi(\tau)+\frac{B}{\sqrt{2\pi}}\int_0^\tau \psi(t)h(\tau-t)dt +\frac{C}{\pi i}\int_\mathbb{R}\frac{\psi(t)}{t-\tau}dt=n(\tau),\N\]\Nwith \(\tau\in(0,+\infty)\). The system of singular integral equations\N\[\N\left\{\begin{array}{ll} \displaystyle A\psi(\tau)+\frac{B}{\sqrt{2\pi}}\int_0^\tau \psi(t)h_1(\tau-t)dt +\frac{C}{\pi i}\int_\mathbb{R}\frac{\psi(t)}{t-\tau}dt=n(\tau), & \tau\in(0,+\infty), \\\N\displaystyle A\psi(\tau)+\frac{B}{\sqrt{2\pi}}\int_\tau^0 \psi(t)h_2(\tau-t)dt +\frac{C}{\pi i}\int_\mathbb{R}\frac{\psi(t)}{t-\tau}dt=n(\tau), & \tau\in(-\infty,0), \end{array} \right.\N\]\Nis considered as well.\NIn all cases, the unknown function \(\psi\) is supposed to belong to \(H_1\).