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Conservative systems of integral convolution equations on the half-line and the whole line. (English. Russian original) Zbl 1062.45002

Sb. Math. 193, No. 6, 847-867 (2002); translation from Mat. Sb. 193, No. 6, 61-82 (2002).
Summary: The following system of convolution integral equations is considered: \[ f(x)=g(x)+\int_a^\infty K(x-t)f(t)dt,\quad -\infty\leq a<\infty, \] where the \((m \times m)\)-matrix-valued function \(K\) satisfies the conditions of conservativeness \[ K_{ij}\in L_1(\mathbb R),\quad K_{ij}\geq 0, \quad A\equiv \int_{\infty}^{\infty} K(x)dx \in P_N, \quad r(A)=1. \] 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 solvability of the inhomogeneous and the homogeneous equations, asymptotic and other properties of solutions are considered. The method of nonlinear factorization equations is applied and developed in combination with new results in multidimensional renewal theory.

MSC:

45F15 Systems of singular linear integral equations
60K05 Renewal theory
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47G10 Integral operators
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