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Renewal equations on the semi-axis. (English. Russian original) Zbl 0937.60083

Izv. Math. 63, No. 1, 57-71 (1999); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 63, No. 1, 61-76 (1999).
Summary: We consider the renewal equation \[ \varphi(x)=g(x)+\int^x_0\varphi(x-t)dF(t),\qquad g\in L_1(0;\infty), \] where \(F\) is the distribution function of a nonnegative random variable. If \(F\) has a nontrivial absolutely continuous component or is a distribution of absolutely continuous type, then we prove that the solution of the renewal equation can be written as follows: \[ \varphi=\varphi_1+\varphi_2+\left[\int^\infty_0 x dF(x)\right]^{-1}\int^\infty_0 g(x) dt, \] where \(\varphi_1\in L_1(0;\infty)\), \(\varphi_2\in C[0;\infty)\), and \(\varphi_2(+\infty)=0\). If \(g\) is bounded and \(g(+\infty)=0\), then \(\varphi_1(+\infty)=0\). The proof is based on the structural factorization of the renewal equation into absolutely continuous, discrete, and singular components.

MSC:

60K05 Renewal theory
45D05 Volterra integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
45M05 Asymptotics of solutions to integral equations
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