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Mat. Sb., 2007, Volume 198, Number 7, Pages 45–62 (Mi msb1483)  

This article is cited in 11 scientific papers (total in 11 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

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.
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English version:
Sbornik: Mathematics, 2007, 198:7, 949–966

Bibliographic databases:

UDC: 517.968.2
MSC: Primary 45E10; Secondary 47G10
Received: 26.12.2005 and 02.10.2006

Citation: L. G. Arabadzhyan, A. S. Khachatryan, “A class of integral equations of convolution type”, Mat. Sb., 198:7 (2007), 45–62; Sb. Math., 198:7 (2007), 949–966

Citation in format AMSBIB
\by L.~G.~Arabadzhyan, A.~S.~Khachatryan
\paper A class of integral equations of convolution type
\jour Mat. Sb.
\yr 2007
\vol 198
\issue 7
\pages 45--62
\jour Sb. Math.
\yr 2007
\vol 198
\issue 7
\pages 949--966

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    This publication is cited in the following articles:
    1. Kh. A. Khachatryan, “O reshenii odnoi sistemy nelineinykh integralnykh uravnenii tipa Gammershteina–Nemytskogo na vsei osi”, Tr. In-ta matem., 21:2 (2013), 154–161  mathnet
    2. Kh. A. Khachatryan, “On positive solutions of one class of nonlinear integral equations of Hammerstein–Nemytskiĭ type on the whole axis”, Trans. Moscow Math. Soc., 75 (2014), 1–12  mathnet  crossref  elib
    3. Kh. A. Khachatryan, “Positive solubility of some classes of non-linear integral equations of Hammerstein type on the semi-axis and on the whole line”, Izv. Math., 79:2 (2015), 411–430  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Kh. A. Khachatryan, T. G. Sardaryan, “O razreshimosti odnogo klassa nelineinykh integralnykh uravnenii tipa Urysona na vsei pryamoi”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 17:1 (2017), 40–50  mathnet  crossref  elib
    5. Khachatryan Kh.A., “On Solvability of One Class of Nonlinear Integral Equations on Whole Line With a Weak Singularity At Zero”, P-Adic Numbers Ultrametric Anal. Appl., 9:4 (2017), 292–305  crossref  mathscinet  zmath  isi  scopus
    6. Kh. A. Khachatryan, H. S. Petrosyan, “One parameter families of positive solutions of some classes of convolution type nonlinear integral equations”, J. Math. Sci., 231:2 (2018), 153–167  mathnet  crossref  crossref
    7. 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
    8. 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
    9. S. M. Andriyan, A. K. Kroyan, Kh. A. Khachatryan, “On solvability of class of nonlinear integral equations in $p$-adic string theory”, Ufa Math. J., 10:4 (2018), 12–23  mathnet  crossref  isi
    10. 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
    11. 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  adsnasa  isi  elib
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