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Izv. RAN. Ser. Mat., 1999, Volume 63, Issue 1, Pages 61–76 (Mi izv228)  

This article is cited in 9 scientific papers (total in 9 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

Full text: PDF file (1024 kB)
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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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\by N.~B.~Engibaryan
\paper Renewal equations on the semi-axis
\jour Izv. RAN. Ser. Mat.
\yr 1999
\vol 63
\issue 1
\pages 61--76
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\crossref{https://doi.org/10.4213/im228}
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\zmath{https://zbmath.org/?q=an:0937.60083}
\transl
\jour Izv. Math.
\yr 1999
\vol 63
\issue 1
\pages 57--71
\crossref{https://doi.org/10.1070/im1999v063n01ABEH000228}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Yengibarian N.B., “Renewal equation on the whole line”, Stochastic Processes and Their Applications, 85:2 (2000), 237–247  crossref  mathscinet  zmath  isi  scopus
    2. M. S. Sgibnev, “Stone decomposition for a matrix renewal measure on a half-line”, Sb. Math., 192:7 (2001), 1025–1033  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. N. B. Engibaryan, “Asymptotic and structural theorems for the Markov renewal equation”, Theory Probab. Appl., 48:1 (2004), 80–92  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. N. B. Engibaryan, “A renewal equation in a multidimensional space”, Theory Probab. Appl., 49:4 (2005), 737–744  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. N. B. Engibaryan, A. Barseghyan, “Random walks and mixtures of Gamma-distributions”, Theory Probab. Appl., 55:3 (2011), 528–535  mathnet  crossref  crossref  mathscinet  isi
    6. Kh. A. Khachatryan, H. S. Petrosyan, “One initial boundary-value problem for integro-differential equation of the second order with power nonlinearity”, Russian Math. (Iz. VUZ), 62:6 (2018), 43–55  mathnet  crossref  isi
    7. 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  mathnet
    8. 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  mathnet  crossref
    9. A. Kh. Khachatryan, Kh. A. Khachatryan, A. S. Petrosyan, “Asimptoticheskoe povedenie resheniya dlya odnogo klassa nelineinykh integro-differentsialnykh uravnenii v zadache raspredeleniya dokhoda”, Tr. IMM UrO RAN, 27, no. 1, 2021, 188–206  mathnet  crossref  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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