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Conservative systems of integral convolution equations
on the half-line and the entire line
N. B. Engibaryan Byurakan Astrophysical Observatory, National Academy of Sciences of Armenia
Abstract:
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.
UDC:
517.9+
519.24
MSC: 45B05,
45D05,
47G10 Received: 11.03.2001
DOI:
10.4213/sm660
Fulltext:
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