Homogeneous Wiener—Hopf Double Integral Equation with Symmetric Kernel in the Conservative Case

Abstract

We establish nontrivial solvability conditions for the homogeneous double integral equation

\(S(x, y)=\int_{0}^{\infty} \int_{0}^{\infty} K\left(x-x^{\prime}, y-y^{\prime}\right) S\left(x^{\prime}, y^{\prime}\right) d x^{\prime} d y^{\prime}, \quad(x, y) \in \mathbb{R}_{+} \times \mathbb{R}_{+},\)

where ℝ₊ ≡ [0, +∞), under the assumption that the given function K satisfies the conservativity conditions

\(0 \leq K \in L_{1}, \quad \iint_{\mathbb{R}^{2}} K(x, y) \ d x\ d y=1\)

and some additional conditions on its first and second moments.