Renewal theorems for a system of integral equations

The system of renewal integral equations
$$ \varphi _i(x)=g_i(x)+\sum _{j=1}^m\int _0^xu_{ij}(x-t)\varphi _j(t) dt, \qquad i=1,…,m, $$
is considered, where the matrix-valued function $u=(u_{ij})$ satisfies the condition of conservativeness $0\leqslant u_{ij}\in L_1^+\equiv L_1(0;\infty)$, and the matrix $A=\int _0^\infty u(x) dx$ is irreducible and of spectral radius.
The existence of a limit at $+\infty$ of the solution $\varphi =(\varphi _1,…,\varphi _m)^T$ is established in the case when the vector-valued function $g=(g_1,…,g_m)^T\in L_1^m$ is bounded and $g(+\infty )=0$. This limit is evaluated. The structure of $\phi$ for $g\in L_1^m$ is determined; namely, $\varphi (x)=\mu +\rho _0(x)+\psi(x)$, where $\rho _0\in C_0^m$ and $\psi \in L_1^m$. A similar formula for the resolvent matrix-valued function is obtained.