A Wiener-Hopf Integral Equation with a Nonsymmetric Kernel in the Supercritical Case

Abstract

The paper is devoted to the solvability questions of the Wiener-Hopf integral equation in the case where the kernel K satisfies the conditions 0 ≤ K ∈ L₁(ℝ), \(\int_{-\infty}^{\infty} K(t)dt>1\), K(±x) ∈ C⁽³⁾(ℝ⁺), (−1)ⁿK(±x)⁽ⁿ⁾(x) ≥ 0, x ∈ ℝ⁺, n =1, 2, 3. Based on Volterra factorization of the Wiener-Hopf operator, and invoking the technique of nonlinear functional equations, we construct real-valued solutions both for homogeneous and non-homogeneous Wiener-Hopf equations, assuming that the function g is real-valued and summable, and the corresponding conditions are satisfied. The behavior at infinity of the corresponding solutions is also studied.