Convolution equations containing singular probability distributions

N. B. Engibaryan

Izvestiya. Mathematics , Tome 60 (1996) no. 2, pp. 251-279

Résumé

The article is devoted to equations of the form \begin{equation} \varphi(x)=g(x)-\int_0^\infty\varphi(t)\,dT(x-t), \tag{1} \end{equation} where $T$ is a continuous function of bounded variation on $(-\infty;\infty)$ containing a singular component. First we study asymptotic and other properties of the solutions of formal Volterra equations (1) corresponding to $T(x)=0$ for $x\leqslant 0$. Next we introduce and study non-linear factorization equations (NFE) for (1). Factorization is constructed in the case when $T(-\infty)=0$, $T(x)\uparrow$ in $x$, and $T(+\infty)=\mu\leqslant 1$. With the aid of this factorization, we prove existence theorems for homogeneous $(g=0)$ and non-homogeneous equations in the singular case $\mu=1$.