A class of integral equations of convolution type

L. G. Arabadzhyan ; A. S. Khachatryan

Sbornik. Mathematics, Tome 198 (2007) no. 7, pp. 949-966

Résumé

Conditions (both necessary and sufficient) for the existence of a non-trivial bounded solution $B$ of the integral equation $$ B(x)=\int_{-\infty}^{+\infty}\lambda(t)K(x-t)B(t)\,dt,\qquad x\in \mathbb R^1, $$ are obtained for fixed functions $K$ and $\lambda$ satisfying the following conditions: \begin{gather*} 0\le K\in L_1(\mathbb R^1), \qquad \int_{-\infty}^\infty K(t)\,dt=1, \\ \int_{-\infty}^\infty t^2K(t)\,dt\infty, \qquad \nu\stackrel{\mathrm{def}}{=}\int_{-\infty}^{+\infty}tK(t)\,dt\ne0, \\ 0\le\lambda(x)\le1, \qquad x\in \mathbb R^1, \qquad \lambda\not\equiv0. \end{gather*} The existence of the limits $B(\pm\infty)=\lim_{x\to\pm\infty}B(x)$ is proved and a relation between these limits, the first-order moment $\nu$, and the integral norm of $B$ is found. Bibliography: 9 titles.