Asymptotic behavior of the solution for one class of nonlinear integral equations of Hammerstein type on the whole axis

Kh. A. Khachatryan ; H. S. Petrosyan

Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 68 (2022) no. 2, pp. 376-391.

Résumé

A class of nonlinear integral equations on the whole axis with a noncompact integral operator of Hammerstein type is investigated. This class of equations has applications in various fields of natural science. In particular, such equations are found in mathematical biology, in the kinetic theory of gases, in the theory of radiation transfer, etc. The existence of a nonnegative nontrivial and bounded solution is proved. The asymptotic behavior of the constructed solution on $\pm\infty$ is studied. In one important special case, the uniqueness of the constructed solution in a certain weighted space is established. At the end of the work, specific applied examples of the equations under study are given.

Bibliographie

[1] Vladimirov V. S., Volovich Ya. I., “O nelineinom uravnenii dinamiki v teorii $p$-adicheskoi struny”, Teor. mat. fiz., 138:3 (2004), 355–368

[2] Engibaryan N. B., “Ob odnoi zadache nelineinogo perenosa izlucheniya”, Astrofizika, 2:1 (1966), 31–36

[3] Zhukovskaya L. V., “Iteratsionnyi metod resheniya nelineinykh integralnykh uravnenii, opisyvayuschikh rollingovye resheniya v teorii strun”, Teor. mat. fiz., 146:3 (2006), 402–409

[4] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1976

[5] Khachatryan A. Kh., Khachatryan Kh. A., Petrosyan A. S., “Asimptoticheskoe povedenie resheniya dlya odnogo klassa nelineinykh integro-differentsialnykh uravnenii v zadache raspredeleniya dokhoda”, Tr. In-ta mat. i mekh. UrO RAN, 27, no. 1, 2021, 188–206

[6] Khachatryan Kh. A., “O razreshimosti nekotorykh klassov nelineinykh integralnykh uravnenii v teorii $p$-adicheskoi struny”, Izv. RAN. Ser. mat., 82:2 (2018), 172–193

[7] Arabadzhyan L. G., “Solutions of certain integral equations of the Hammerstein type”, J. Contemp. Math. Anal., 32:1 (1997), 17–24

[8] Arabadzhyan L. G., Khachatryan A. S., “A class of integral equations of convolution type”, Sb. Math., 198:7 (2007), 949–966

[9] Barbour A. D., “The uniqueness of Atkinson and Reuter's epidemic waves”, Math. Proc. Cambridge Phil. Soc., 82:1 (1977), 127–130

[10] Cercignani C., The Boltzmann Equation and Applications, Springer, New York, 1988

[11] Diekmann O., “Thresholds and travelling waves for the geographical spread of infection”, J. Math. Biol., 6:2 (1978), 109–130

[12] Khachatryan A. Kh., Khachatryan Kh. A., “Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave”, Theoret. and Math. Phys., 189:2 (2016), 1609–1623

[13] Khachatryan A. Kh., Khachatryan Kh. A., “On the solvability of some nonlinear integral equations in problems of epidemic spread”, Proc. Steklov Inst. Math., 306 (2019), 271–287

[14] Khachatryan Kh. A., “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

[15] Khachatryan Kh. A., Petrosyan H. S., “On the solvability of a class of nonlinear Hammerstein–Stieltjes integral equations on the whole line”, Proc. Steklov Inst. Math., 308 (2020), 238–249

[16] Khachatryan Kh. A., Petrosyan H. S., “Some integral equations on the whole line with monotone nonlinearity and a difference kernel”, J. Math. Sci. (N. Y.), 255:6 (2021), 790–804