Multidimensional renewal equations and moments of branching processes

We mean by a multidimensional renewal equation a system of equations
\begin{gather*} X^l_m(x)=K^l_m(x)+\sum_{\alpha=1}^n\int_0^xX^\alpha_m(x-u)\,dF^l_\alpha(u) \\ l=1,\dots,n;\quad m=1,\dots,N, \end{gather*}
where $F^l_m(x)$ are non-decreasing right-continious non-negative functions, $F^l_m(0)=0$, ($l,m=1,\dots,n$) and $K^l_m(x)$, $l,=1,\dots,n$, $m=1,\dots,N$ are measurable bounded functions satisfying some conditions. The asymptotic behaviour of solution $X^l_m(x)$ is described in theorems 2.1–2.7. We use these theorems to investigate asymptotic behaviour of the first and second moments of age-dependent branching processes with $n$ types of particles.