On a convolution-type integral equation in unbounded planar domain (Q1382899)

The author considers the two-dimensional Wiener-Hopf equation \[ f(x,y)= g(x,y)+\iint_G K(x-x', y-y')f(y,y') dx' dy',\tag{1} \] Here \(G\in\mathbb{R}^2\) is a domain with piecewise smooth boundary such that \(\Pi_{\Gamma_1}\cap G \cap \Pi_{\Gamma_2}\) and \(\Pi_{\Gamma_j}, j=1,2\) are half-planes and the kernel \(K(x,y)\) satisfies the condition \(K(x,y) \geq 0\) and \(\mu=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}K(x,y) dx dy \leq 1\). The author gives a condition for the existence of a positive solution of the homogeneous equation (1).