Email this article On the solvability of a class of nonlinear two-dimensional integral equations Hammerstein–Nemytskii type on the plane
- Authors: Khachatryan K.A.1, Petrosyan H.S.2
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Affiliations:
- Yerevan State University
- Armenian National Agrarian University
- Issue: Vol 27, No 3 (2023)
- Pages: 446-461
- Section: Differential Equations and Mathematical Physics
- URL: https://journal-vniispk.ru/1991-8615/article/view/310972
- DOI: https://doi.org/10.14498/vsgtu2013
- ID: 310972
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Abstract
We consider a class of nonlinear integral equations with a stochastic and symmetric kernel on the whole line. With certain particular representations of the kernel and nonlinearity, equations of the above character arise in many branches of mathematical natural science. In particular, such equations occur in the theory p-adic strings, in the kinetic theory of gases, in mathematical biology and in the theory of radiative transfer. Constructive existence theorems are proved for non-negative non-trivial and bounded solutions under various restrictions on the function describing the nonlinearity in the equation. Under additional restrictions on the kernel and on the nonlinearity, a uniqueness theorem is also proved in a certain class of bounded and non-negative functions that have a finite limit in ±∞. Specific applied examples of the kernel and non-linearity are given that satisfy all the restrictions of the proven statements.
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##article.viewOnOriginalSite##About the authors
Khachatur A. Khachatryan
Yerevan State University
Author for correspondence.
Email: khachatur.khachatryan@ysu.am
ORCID iD: 0000-0002-4835-943X
http://www.mathnet.ru/person27540
D.Sc. (Phys. & Math. Sci.), Professor; Head of the Dept., Dept. of Theory of Functions and Differential Equations
Armenia, 0025, Yerevan, A. Manukyan str., 1Haykanush S. Petrosyan
Armenian National Agrarian University
Email: haykuhi25@mail.ru
ORCID iD: 0000-0002-7172-4730
Cand. Phys. & Math. Sci., Associate Professor, Dept of Higher Mathematics and Physics
Armenia, 0009, Yerevan, Marshal Teryan str., 74References
- Volovich I. V. $p$-adic string, Class. Quantum Grav., 1987, vol. 4, no. 4, pp. L83–L87. DOI: https://doi.org/10.1088/0264-9381/4/4/003.
- Moeller N., Schnabl M. Tachyon condensation in open-closed $p$-adic string theory, J. High Energ. Phys., 2004, vol. 2004, no. 01, 011, arXiv: hep-th/0304213. DOI: https://doi.org/10.1088/1126-6708/2004/01/011.
- Vladimirov V. S. Nonlinear equations for 𝑝-adic open, closed, and open-closed strings, Theoret. and Math. Phys., 2006, vol. 149, no. 3, pp. 1604–1616. EDN: LJOAVJ. DOI: https://doi.org/10.1007/s11232-006-0144-z.
- Diekmann O. Thresholds and travelling waves for the geographical spread of infection, J. Math. Biology, 1978, vol. 6, no. 2, pp. 109–130. DOI: https://doi.org/10.1007/BF02450783.
- Diekmann O., Kaper H. G. On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., Theory Methods Appl., 1978, vol. 2, no. 6, pp. 721–737. DOI: https://doi.org/10.1016/0362-546X(78)90015-9.
- Cercignani C. The Boltzmann Equation and Applications, Applied Mathematical Sciences, vol. 67. New York, Springer, 1988, 455 pp. DOI: https://doi.org/10.1007/978-1-4612-1039-9.
- Aref'eva I. Ya., Volovich I. V. Cosmological daemon, J. High Energ. Phys., 2011, vol. 2011, no. 8, 102, arXiv: 1103.0273 [hep-th]. DOI: https://doi.org/10.1007/JHEP08(2011)102.
- Khachatryan Kh. A., Petrosyan H. S., Avetisyan M. H. Solvability issues for a class of convolution type nonlinear integral equations in R𝑛, Trudy Inst. Mat. Mekh. UrO RAN, 2018, vol. 24, no. 3, pp. 247–262 (In Russian). EDN: UXZCME. DOI: https://doi.org/10.21538/0134-4889-2018-24-3-247-262.
- Khachatryan Kh. A., Petrosyan H. S. Alternating bounded solutions of a class of nonlinear two-dimensional convolution-type integral equations, Trans. Moscow Math. Soc., 2021, vol. 82, pp. 259–271. EDN: JDJCFK. DOI: https://doi.org/10.1090/mosc/329.
- Khachatryan Kh. A., Petrosyan H. S. On the construction of an integrable solution to one class of nonlinear integral equations of Hammerstein–Nemytskii type on the whole axis, Trudy Inst. Mat. Mekh. UrO RAN, 2020, vol. 26, no. 2, pp. 278–287 (In Russian). EDN: UTSHCD. DOI: https://doi.org/10.21538/0134-4889-2020-26-2-278-287.
- Khachatryan A. Kh., Khachatryan Kh. A. Hammerstein–Nemytskii type nonlinear integral equations on half-line in space $L_1(0,+infty) cap L_{infty}(0,+infty)$, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math., 2013, vol. 52, no. 1, pp. 89–100.
- 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. Equat., 2021, vol. 57, no. 6, pp. 768–779. EDN: UVRUYT. DOI: https://doi.org/10.1134/S0012266121060069.
- Arabadzhyan L. G. Existence of nontrivial solutions of certain linear and nonlinear convolution-type equations, Ukr. Math. J., 1989, vol. 41, no. 12, pp. 1359-1367. DOI: https://doi.org/10.1007/BF01056100.
- Kolmogorov A. N., Fomin S. V. Elementy teorii funktsii i funktsional’nogo analiza Elements of the Theory of Functions and Functional Analysis. Moscow, Nauka, 1981, 542 pp. (In Russian)
- Arabadzhyan L. G., Khachatryan A. S. A class of integral equations of convolution type, Sb. Math., 2007, vol. 198, no. 7, pp. 949–966. EDN: XLRESL. DOI: https://doi.org/10.1070/SM2007v198n07ABEH003868.
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