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Sbornik: Mathematics, 2007, Volume 198, Issue 7, Pages 949–966
DOI: https://doi.org/10.1070/SM2007v198n07ABEH003868
(Mi sm1483)
 

This article is cited in 29 scientific papers (total in 29 papers)

A class of integral equations of convolution type

L. G. Arabadzhyanab, A. S. Khachatryanb

a Institute of Mathematics, National Academy of Sciences of Armenia
b Armenian State Teachers' Training University named after Khachatur Abovian
References:
Abstract: Conditions (both necessary and sufficient) for the existence of a non-trivial bounded solution $B$ of the integral equation
$$ B(x)=\int_{-\infty}^{+\infty}\lambda(t)K(x-t)B(t)\,dt,\qquad x\in \mathbb R^1, $$
are obtained for fixed functions $K$ and $\lambda$ satisfying the following conditions:
\begin{gather*} 0\le K\in L_1(\mathbb R^1), \qquad \int_{-\infty}^\infty K(t)\,dt=1, \\ \int_{-\infty}^\infty t^2K(t)\,dt<\infty, \qquad \nu\stackrel{\mathrm{def}}{=}\int_{-\infty}^{+\infty}tK(t)\,dt\ne0, \\ 0\le\lambda(x)\le1, \qquad x\in \mathbb R^1, \qquad \lambda\not\equiv0. \end{gather*}
The existence of the limits $B(\pm\infty)=\lim_{x\to\pm\infty}B(x)$ is proved and a relation between these limits, the first-order moment $\nu$, and the integral norm of $B$ is found.
Bibliography: 9 titles.
Received: 26.12.2005 and 02.10.2006
Bibliographic databases:
UDC: 517.968.2
MSC: Primary 45E10; Secondary 47G10
Language: English
Original paper language: Russian
Citation: L. G. Arabadzhyan, A. S. Khachatryan, “A class of integral equations of convolution type”, Sb. Math., 198:7 (2007), 949–966
Citation in format AMSBIB
\Bibitem{AraKha07}
\by L.~G.~Arabadzhyan, A.~S.~Khachatryan
\paper A class of integral equations of convolution type
\jour Sb. Math.
\yr 2007
\vol 198
\issue 7
\pages 949--966
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Linking options:
  • https://www.mathnet.ru/eng/sm1483
  • https://doi.org/10.1070/SM2007v198n07ABEH003868
  • https://www.mathnet.ru/eng/sm/v198/i7/p45
  • This publication is cited in the following 29 articles:
    1. Kh. A. Khachatryan, H. S. Petrosyan, “Asymptotic Behavior of the Solution for One Class of Nonlinear Integral Equations of Hammerstein Type on the Whole Axis”, J Math Sci, 282:2 (2024), 292  crossref
    2. Kh. A. Khachatryan, H. S. Petrosyan, “On the Solvability of One Infinite System of Integral Equations with Power Nonlinearity on the Semi-Axis”, J. Contemp. Mathemat. Anal., 59:4 (2024), 305  crossref
    3. Zahra Keyshams, Khachatur A. Khachatryan, Monire Mikaeili Nia, “Existence and Uniqueness Theorems for One Class of Hammerstein-type Nonlinear Integral Equations”, Lobachevskii J Math, 45:8 (2024), 3580  crossref
    4. Kh. A. Khachatryan, A. S. Petrosyan, M. O. Avetisyan, “Teoremy suschestvovaniya i edinstvennosti dlya odnoi sistemy integralnykh uravnenii s dvumya nelineinostyami”, Tr. IMM UrO RAN, 29, no. 1, 2023, 202–218  mathnet  crossref  mathscinet  elib
    5. A. Kh. Khachatryan, Kh. A. Khachatryan, H. S. Petrosyan, “On nonlinear convolution-type integral equations in the theory of $p$-adic strings”, Theoret. and Math. Phys., 216:1 (2023), 1068–1081  mathnet  crossref  crossref  mathscinet  adsnasa
    6. Kh. A. Khachatryan, H. S. Petrosyan, “On non-trivial solvability of one system of non-linear integral equations on the real axis”, Izv. Math., 87:5 (2023), 1062–1077  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    7. Kh. A. Khachatryan, A. S. Petrosyan, “O razreshimosti odnogo klassa nelineinykh dvumernykh integralnykh uravnenii tipa Gammershteina–Nemytskogo na ploskosti”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 27:3 (2023), 446–461  mathnet  crossref
    8. A. A. Davydov, Kh. A. Khachatryan, A. S. Petrosyan, “On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line”, Differentsialnye uravneniya, 59:11 (2023), 1500  crossref
    9. A. A. Davydov, Kh. A. Khachatryan, H. S. Petrosyan, “On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line”, Diff Equat, 59:11 (2023), 1504  crossref
    10. Kh. A. Khachatryan, A. S. Petrosyan, “Asimptoticheskoe povedenie resheniya dlya odnogo klassa nelineinykh integralnykh uravnenii tipa Gammershteina na vsei pryamoi”, SMFN, 68, no. 2, Rossiiskii universitet druzhby narodov, M., 2022, 376–391  mathnet  crossref  mathscinet
    11. Kh. A. Khachatryan, H. S. Petrosyan, “On a class of convolution type nonlinear integral equations with an even kernel”, Int. J. Mod. Phys. A, 37:20n21 (2022)  crossref
    12. Khachatryan A.Kh., Khachatryan Kh.A., Petrosyan H.S., “On Positive Bounded Solutions of One Class of Nonlinear Integral Equations With the Hammerstein-Nemytskii Operator”, Differ. Equ., 57:6 (2021), 768–779  crossref  mathscinet  isi
    13. Kh. A. Khachatryan, A. S. Petrosyan, “O polozhitelnykh resheniyakh granichnoi zadachi dlya nelineinogo integro-differentsialnogo uravneniya na polubeskonechnom intervale”, Vladikavk. matem. zhurn., 22:2 (2020), 70–82  mathnet  crossref
    14. Kh. A. Khachatryan, A. S. Petrosyan, “O znakoperemennykh i ogranichennykh resheniyakh odnogo klassa integralnykh uravnenii na vsei osi s monotonnoi nelineinostyu”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 24:4 (2020), 644–662  mathnet  crossref
    15. Khachatryan Kh.A., Petrosyan H.S., “Solvability of a Nonlinear Problem in Open-Closed P-Adic String Theory”, Differ. Equ., 56:10 (2020), 1371–1378  crossref  mathscinet  isi  scopus
    16. Khachatryan Kh.A., Andriyan S.M., “On Solvability of One Class of Nonlinear Integral Equations on Whole Line With Two Monotone Nonlinearities”, P-Adic Numbers Ultrametric Anal. Appl., 12:4 (2020), 259–275  crossref  mathscinet  isi
    17. Khachatryan Kh.A., Andriyan S.M., “On the Solvability of a Class of Discrete Matrix Equations With Cubic Nonlinearity”, Ukr. Math. J., 71:12 (2020), 1910–1928  crossref  mathscinet  isi
    18. Kh. A. Khachatryan, “Solvability of some nonlinear boundary value problems for singular integral equations of convolution type”, Trans. Moscow Math. Soc., 81:1 (2020), 1–31  mathnet  crossref  elib
    19. Kh. A. Khachatryan, “Solvability of some classes of nonlinear singular boundary value problems in the theory of $p$-adic open–closed strings”, Theoret. and Math. Phys., 200:1 (2016), 1015–1025  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    20. Khachatryan Kh.A., Terdzhyan Ts.E., Sardanyan T.G., “On the Solvability of One System of Nonlinear Hammerstein-Type Integral Equations on the Semiaxis”, Ukr. Math. J., 69:8 (2018), 1287–1305  crossref  mathscinet  zmath  isi  scopus
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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