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This article is cited in 8 scientific papers (total in 8 papers)
Renewal equations on the semi-axis
N. B. Engibaryan Byurakan Astrophysical Observatory, National Academy of Sciences of Armenia
Abstract:
We consider the renewal equation
$$
\varphi(x)=g(x)+\int_0^x\varphi(x-t) dF(t), \qquad g\in L_1(0;\infty),
$$
where $F$ is the distribution function of a non-negative random variable. If $F$ has a non-trivial 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:
$$
\varphi=\varphi_1+\varphi_2+[\int_0^{\infty}x dF(x)]^{-1}\int_0^{\infty}g(x) dt,
$$
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.
DOI:
https://doi.org/10.4213/im228
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English version:
Izvestiya: Mathematics, 1999, 63:1, 57–71
Bibliographic databases:
MSC: 60K05, 45D05, 45E10, 47B35, 47A68, 45M05 Received: 18.02.1997
Citation:
N. B. Engibaryan, “Renewal equations on the semi-axis”, Izv. RAN. Ser. Mat., 63:1 (1999), 61–76; Izv. Math., 63:1 (1999), 57–71
Citation in format AMSBIB
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http://mi.mathnet.ru/eng/izv228https://doi.org/10.4213/im228 http://mi.mathnet.ru/eng/izv/v63/i1/p61
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N. B. Engibaryan, “Asymptotic and structural theorems for the Markov renewal equation”, Theory Probab. Appl., 48:1 (2004), 80–92
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N. B. Engibaryan, “A renewal equation in a multidimensional space”, Theory Probab. Appl., 49:4 (2005), 737–744
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N. B. Engibaryan, A. Barseghyan, “Random walks and mixtures of Gamma-distributions”, Theory Probab. Appl., 55:3 (2011), 528–535
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Kh. A. Khachatryan, S. M. Andriyan, A. A. Sisakyan, “On the solvability of a class of boundary value problems for systems of the integral equations with power nonlinearity on the whole axis”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2018, no. 2, 54–73
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Kh. A. Khachatryan, A. K. Kroyan, “Suschestvovanie nechetnogo resheniya dlya odnoi granichnoi zadachi so stepennoi nelineinostyu”, Sib. zhurn. chist. i prikl. matem., 18:4 (2018), 88–96
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