The Wiener--Hopf Integral Equation in the Supercritical Case

L. G. Arabadzhyan

Matematičeskie zametki, Tome 76 (2004) no. 1, pp. 11-19

Résumé

We consider the scalar homogeneous equation $$ S(x)=\int_0^\infty K(x-t)S(t)\,dt, \qquad x\in\mathbb R^+\equiv(0,\infty), $$ with symmetric kernel $K$: $K(-x)=K(x)$, $x\in\mathbb R_1$ satisfying the conditions $$ 0\leqslant K\in L_1(\mathbb R^+)\cap C^{(2)}(\mathbb R^+), \qquad \int_0^\infty K(t)\,dt>\frac12, $$ $K'\leqslant 0$ and $0\leqslant K''\downarrow$ on $\mathbb R^+$. We prove the existence of a real solution $S$ of the equation given above with asymptotic behavior $S(x)=O(x)$ as $x\to+\infty$.