zbMATH — the first resource for mathematics

The Wiener-Hopf integral equation in the supercritical case. (English. Russian original) Zbl 1058.45001
Math. Notes 76, No. 1, 10-17 (2004); translation from Mat. Zametki 76, No. 1, 11-19 (2004).
The author studies an asymptotic behavior of the solutions of the homogeneous Wiener-Hopf integral equation \(S(x)=\int_0^\infty K(x-t)S(t) \,dt\), \(x>0,\) with smooth nonnegative even kernel \(K(x)\) such that \(K^{\prime}(x)\leq 0, K^{\prime\prime}(x)\geq 0,\) and \(K^{\prime\prime}(x) \downarrow \) on \(R^+\). It is proved that in the so-called supercritical case \(\int_0^\infty K(x) \,dx >1/2\) there exists a real solution \(S(x)\) such that \(S(x)=O(x)\) as \(x\to\infty.\) The consideration is based on the factorization of the Wiener-Hopf operator which coincides with the solution problem for the nonlinear integral equation \(V(x)= K(x)+ \int_0^\infty V(t)V(x+t)\,dt, x>0\).

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
45G05 Singular nonlinear integral equations
Full Text: DOI