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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 $$V(x)= K(x)+ \int_0^\infty V(t)V(x+t)\,dt, x>0$$.

##### MSC:
 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
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