Kh. A. Khachatryan, H. S. Petrosyan. On the construction of an integrable solution to one class of nonlinear integral equations of Hammerstein–Nemytskii type on the whole axis ... P. 278-287

We study one class of nonlinear integral equations of convolution type with the Hammerstein–Nemytskii operator on the whole axis. This class has direct applications in the kinetic theory of gases, the theory of p-adic open-closed strings, and the theory of radiative transfer. We prove a constructive theorem on the existence of a nontrivial nonnegative solution integrable on the whole axis. In the end of the paper, we give specific examples of such equations satisfying all conditions of the main theorem.

Keywords: Hammerstein–Nemytskii equations, successive approximations, monotonicity, convexity, convergence of iterations.

REFERENCES

1.   Cercignani C. The Boltzmann equation and its applications. N Y: Springer, 1988. 455 p.
doi: 10.1007/978-1-4612-1039-9 .

2.   Zel’dovich Ya.B., Raizer Yu.P. Physics of shock waves and high-temperature hydrodynamic phenomena. Mineola: Dover, 2002, 916 p. ISBN: 9780486420028 . Original Russian text published in Zel’dovich Ya.B., Raizer Yu.P. Fizika udarnykh voln i vysokotemperaturnykh gidrodinamicheskikh yavlenii. Moscow: Nauka Publ., 1966.

3.   Vladimirov V.S., Volovich Ya.I. Nonlinear dynamics equation in p-adic string theory. Theoretical and Mathematical Physics, 2004, vol. 138, no. 3, pp. 297–309. doi: 10.1023/B:TAMP.0000018447.02723.29 .

4.   Khachatryan Kh.A. On the solvability of certain classes of non-linear integral equations in p-adic string theory. Izv. Math., 2018, vol. 82, no. 2, pp. 407–427. doi: 10.1070/IM8580 .

5.   Engibaryan N.B. On a problem in non-linear radiative transfer. Astrophysics, 1966, vol. 2, no. 1, pp. 12–14. doi: 10.1007/BF01014505 .

6.   Khachatryan Kh.A. On positive solutions of one class of nonlinear integral equations of Hammerstein–Nemytskii type on the whole axis. Trans. Moscow Math. Soc., 2014, vol. 75, no. 1, pp. 1–12. doi: 10.1090/S0077-1554-2014-00226-0 .

7.   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., 2015, vol. 79, no. 2, pp. 411–430. doi: 10.1070/IM2015v079n02ABEH002748 .

8.   Khachatryan Kh.A. On solution of a system of Hammerstein–Nemytskii type non-linear integral equations on whole axis. Tr. Inst. Mat., 2013, vol. 21, no. 2, pp. 154–161 (in Russian).

9.   Diekmann O. Thresholds and travelling waves for the geographical spread of infection. J. Math. Biology, 1978, vol. 6, no. 2, pp. 109—130.

10.   Vladimirov V.S. Nonlinear equations for p-adic open, closed and open–closed strings. Theoret. and Mathem. Physics, 2006, vol. 149, no. 3, pp. 1604–1616. doi: 10.1007/s11232-006-0144-z .

11.   Gomez C., Prado H., Trofimchuk S. Separation dichotomy and wavefronts for a nonlinear convolution equation. J. Math. Anal. Appl., 2014, vol. 420, pp. 1—19. doi: 10.1016/j.jmaa.2014.05.064 .

12.   Zhukovskaya L.V. Iterative method for solving nonlinear integral equations describing rolling solutions in string theory. Theoret. and Math. Phys., 2006, vol. 146, no. 3, pp. 335–342. doi: 10.1007/s11232-006-0043-3 .

13.   Khachatryan Kh.A. On the solvability of a boundary-value problem in p-adic string theory. Trans. Moscow Math. Soc., 2018, pp. 101–115.

14.   Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis (Two volumes in one, translated from the first Russian edition 1957–1961). United States: Martino Fine Books. 2012, 280 p. ISBN: 1614273049 . The 7th edition of Russian text published in Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional’nogo analiza. Moscow: Nauka Publ., 2004, 572 p.

15.   Rudin W. Functional Analysis. N Y: McGraw-Hill, 1973, 397 p. ISBN: 978-0070542259 . Translated to Russian under the title Funktsional’nyi analiz. Moscow: Mir Publ., 1975, 499 p.

Received November 18, 2019

Revised January 22, 2020

Accepted January 27, 2020

Funding Agency: The study was funded by a grant from the Russian Science Foundation (project № 19-11-00223).

Khachatur Aghavardovich Khachatryan, Dr. Phys.-Math. Sci., Institute of Mathematics, National Academy of Sciences of Armenia, 0019, Yerevan, Republic of Armenia; Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 119991 Russia,
e-mail: Khach82@rambler.ru.

Haykanush Samvelovna Petrosyan, Cand. Sci. (Phys.-Math.), Armenian National Agrarian University, 0009, Yerevan, Republic of Armenia; Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 119991 Russia, e-mail: Haykuhi25@mail.ru.

Cite this article as: Kh.A.Khachatryan, H.S.Petrosyan. On the construction of an integrable solution to one class of nonlinear integral equations of Hammerstein–Nemytskii type on the whole axis. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 2, pp. 278–287.