On bounded solutions of a class of nonlinear integral equations on the plane and the Urysohn equation in a quadrant of the plane

  • Kh. A. Khachatryan Yerevan. state University, Institute of Mathematics of the National Academy of Sciences of Armenia, Moscow. state un-t im. MV Lomonosov, Russia
  • H. S. Petrosyan Nat. agrarian. University of Armenia, Moscow. state un-t im. MV Lomonosov, Russia

Abstract

UDC 517.968.4

We study a class of two-dimensional integral equations in the plane with monotonic nonlinearity.
These equations have a lot of applications in many fields of natural science.
For example, such equations arise in the dynamic theory of $p$-adic open-closed strings, in the mathematical theory of spatio-temporal spread of epidemics, in the kinetic theory of gases (the Boltzmann kinetic equation in the framework of various models), in the theory of radiative transfer.

We prove a constructive existence theorem for bounded nontrivial solutions and for solutions with alternating sign.
It is shown that obtained results have applications in the theory of $p$-adic open-closed strings and in mathematical biology.
The methods used in the proof of the theorem make it possible to investigate a class of two-dimensional integral equations of the Urysohn type in a quadrant of the plane.
At the end of the paper, we provide specific examples of applications of these equations to illustrate the obtained results.

References

V. S. Vladimirov, Ya. I. Volovich, O nelinejnom uravnenii dinamiki v teorii p-adicheskoj struny`, Teor. i mat. fizika,138, № 3, 355 – 368 (2004). DOI: https://doi.org/10.4213/tmf36

V. S. Vladimirov, O nelinejny`kh uravneniyakh p-adicheskikh otkry`ty`kh, zamknuty`kh i otkry`to-zamknuty`kh strun, Teor. i mat. fizika, 149, № 3, 354 – 367 (2006).

Kh. A. Khachatryan, O razreshimosti nekotory`kh klassov nelinejny`kh singulyarny`kh kraevy`kh zadach, voznikayushhikh v teorii $p$-adicheskikh otkry`to-zamknuty`kh strun, Teor. i mat. fizika 200, № 1, 106 – 117 (2019).

I. Ya. Arefeva, B. G. Dragovic, I. V. Volovich, Open and closed p-adic strings and quadratic extensions of number fields, Phys. Lett. B, 212, № 3, 283 – 291 (1988), https://doi.org/10.1016/0370-2693(88)91318-4 DOI: https://doi.org/10.1016/0370-2693(88)91318-4

O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biology, 6, № 2, 109 – 130 (1978), https://doi.org/10.1007/BF02450783 DOI: https://doi.org/10.1007/BF02450783

A. G. Sergeev, Kh. A. Khachatryan, O razreshimosti odnogo klassa nelinejny`kh integral`ny`kh uravnenij v zadache rasprostraneniya e`pidemii, Tr. Mosk. mat. o-va, 80, № 1, 113 – 131 (2019).

C. Cercignani, The Boltzmann equation and its applications, Springer-Verlag, New York (1988), https://doi.org/10.1007/978-1-4612-1039-9 DOI: https://doi.org/10.1007/978-1-4612-1039-9_2

N. B. Engibaryan, Ob odnoj zadache nelinejnogo perenosa izlucheniya, Astrofizika, 2, № 1, 31 – 36 (1966).

Kh. A. Khachatryan, O razreshimosti odnoj granichnoj zadachi v $p$-adicheskoj teorii strun, Tr. Mosk. mat. o-va, 79, № 1, 117 – 132 (2018).

Kh. A. Khachatryan, Sushhestvovanie i edinstvennost` resheniya odnoj granichnoj zadachi dlya integral`nogo uravneniya svertki s monotonnoj nelinejnost`yu, Izv. RAN. Ser. mat., 84, № 4, 198 – 207 (2020). DOI: https://doi.org/10.4213/im8898

S. M. Andrian, A. K. Kroyan, Kh. A. Khachatryan, O razreshimosti odnogo klassa nelinejny`kh integral`ny`kh uravnenij v p-adicheskoj teorii strun, Ufim. mat. zhurn.,10, № 4, 12 – 23 (2018).

Kh. A. Khachatryan, O razreshimosti nelinejny`kh granichny`kh zadach dlya singulyarny`kh integral`ny`kh uravnenij tipa svertki, Tr. Mosk. mat. o-va, 81, № 1, 3 – 40 (2020).

Kh. A. Khachatryan, A. S. Petrosyan, M. O. Avetisyan, Voprosy` razreshimosti odnogo klassa nelinejny`kh integral`ny`kh uravnenij tipa svertki v $R_n$, Tr. In-ta matematiki i mekhaniki UrO RAN, 24, № 3, 247 – 262 (2018).

L. G. Arabadzhyan, N. B. Engibaryan, Uravneniya v svertkakh i nelinejny`e funkczional`ny`e uravneniya, Itogi nauki i tekhniki. Ser. Mat. analiz, 22, 175 – 244 (1984).

U. Rudin, Funkczional`ny`j analiz, Mir, Moskva (1975).

A. N. Kolmogorov, S. V. Fomin, Элементы теории функций и функционального анализа (Russian) [[Elements of the theory of functions and functional analysis]], V-е isd., ``Nauka'', Moscow, (1981).

M. A. Krasnosel`skij, P. P. Zabrejko i dr., Integral`ny`e operatory` v prostranstvakh summiruemy`kh funkczij, ``Nauka'', Moscow, (1966).

Kh. A. Khachatryan, O razreshimosti odnogo klassa dvumerny`kh integral`ny`kh uravnenij Ury`sona na chetverti ploskosti, Mat. tr., 20, № 2, 193 – 205 (2017).

Published
24.05.2021
How to Cite
Khachatryan , K. A., and H. S. Petrosyan. “On Bounded Solutions of a Class of Nonlinear Integral Equations on the Plane and the Urysohn Equation in a Quadrant of the Plane”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 5, May 2021, pp. 695 -11, doi:10.37863/umzh.v73i5.6541.
Section
Research articles