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.
\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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\crossref{https://doi.org/10.1070/SM2007v198n07ABEH003868}
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This publication is cited in the following 29 articles:
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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