Conservative systems of integral convolution equations

N. B. Engibaryan

Sbornik. Mathematics, Tome 193 (2002) no. 6, pp. 847-867

Résumé

The following system of integral convolution equations is considered: $$ f(x)=g(x)+\int_a^\infty K(x-t)f(t)\,dt, \qquad -\infty\leqslant a\infty, $$ where the $(m\times m)$-matrix-valued function $K$ satisfies the conditions of conservativeness $$ K_{ij}\in L_1(\mathbb R), \quad K_{ij}\geqslant 0, \qquad A\equiv\int_{-\infty}^\infty K(x)\,dx\in P_N, \qquad r(A)=1. $$ Here $P_N$ is the class of non-negative indecomposable $(m\times m)$-matrices and $r(A)$ is the spectral radius of the matrix $A$. For $a=0$ the equation in question is a conservative system of Wiener–Hopf integral equations. For $a=-\infty$ this is the multidimensional renewal equation on the entire line. Questions of the solubility of the inhomogeneous and the homogeneous equations, asymptotic and other properties of solutions are considered. The method of non-linear factorization equations is applied and developed in combination with new results in multidimensional renewal theory.