The Wiener–Hopf Integral Equation in the Supercritical Case

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 $x\to+\infty$.