Solvability of some nonlinear boundary value problems for singular integral equations of convolution type

Kh. A. Khachatryan

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 81 (2020) no. 1, pp. 3-40.

Résumé

This work is devoted to questions about the existence and uniqueness of solutions to certain nonlinear boundary value problems for singular integral equations of convolution type on the whole straight line. It also looks at their asymptotic properties. Several particular cases of this problem have direct applications in $ p$-adic string theory, the mathematical theory of the geographic spread of epidemics, the kinetic theory of gases and radiative transfer theory. For the two classes of boundary value problems described by such equations, the existence of a nontrivial bounded continuous solution is proved, and the asymptotics of the solution that is constructed are investigated. In certain classes of functions which are bounded and continuous on the whole numerical axis, it is shown that no more than one solution exists. The results obtained are extended to certain nonlinear Urysohn-type equations and to Hammerstein-type equations with two nonlinearities. It is also proved that, in certain special cases, solutions to equations having a continuous convex nonlinearity have a series of important properties. Examples of applications of the above equations are given which illustrate the features of the results obtained.

Bibliographie

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

[2] N. B. Engibaryan, A. Kh. Khachatryan, “O tochnoi linearizatsii zadach skolzheniya razrezhennogo gaza v modeli Bkhatnagara-Grossa-Kruka”, TMF, 125:2 (2000), 339–342 | MR | Zbl

[3] A. Kh. Khachatryan, Kh. A. Khachatryan, “Kachestvennye razlichiya reshenii dlya odnoi modeli uravneniya Boltsmana v lineinom i nelineinom sluchayakh”, TMF, 172:3 (2012), 497–504 | Zbl

[4] A. Kh. Khachatryan, Kh. A. Khachatryan, “O razreshimosti nelineinogo modelnogo uravneniya Boltsmana v zadache ploskoi udarnoi volny”, TMF, 189:2 (2016), 239–255 | MR | Zbl

[5] J. J. Keller, “Propagation of simple non-linear waves in gas filled tubes with friction”, Z. angew. Math. Phys., 32:2 (1981), 170–181 | DOI | Zbl

[6] O. Diekmann, “Thresholds and traveling waves for the geographical spread of infection”, J. Math. Biol., 6 (1978), 109–130 | DOI | MR | Zbl

[7] O. Diekmann, “Run for your life. A note on the asymptotic speed of propagation of an epidemic”, J. Dif. Equations, 33:1 (1979), 58–73 | DOI | MR | Zbl

[8] A. Kh. Khachatryan, Kh. A. Khachatryan, “O razreshimosti nekotorykh nelineinykh integralnykh uravnenii v zadachakh rasprostraneniya epidemii”, Matematicheskaya fizika i prilozheniya, Sbornik statei. K 95-letiyu so dnya rozhdeniya akademika V. S. Vladimirova, Tr. MIAN, 306, MIAN, M., 2019, 287–303

[9] I. Ya. Aref'eva, B. G. Dragović, I. V. Volovich, “Open and closed $p$-adic strings and quadratic extensions of number fields”, Phys. Lett. B, 212:3 (1988), 283–291 | DOI | MR

[10] I. Ya. Aref'eva, A. S. Koshelev, L. V. Joukovskaya, “Time evolution in superstring field theory on non-BPS brane. I. Rolling tachyon and energy-momentum conservation”, J. High Energy Phys., 2003, no. 9, 012 | DOI | MR

[11] I. Ya. Arefeva, “Skatyvayuschiesya resheniya polevykh uravnenii na neekstremalnykh branakh i v $p$-adicheskikh strunakh”, Izbrannye voprosy $p$-adicheskoi matematicheskoi fiziki i analiza, Sbornik statei. K 80-letiyu so dnya rozhdeniya akademika V. S. Vladimirova, Tr. MIAN, 245, Nauka, M., 2004, 47–54

[12] I. Ya. Arefeva, “Puzzles with Tachyon in SSFT and Cosmological Applications”, Prog. Theor. Phys., 188, Suppl. (2011), 29–40 | DOI | Zbl

[13] K. Ohmori, “Toward an open-closed string theoretical description of arolling tachyon”, Phys. Rev. D, 69 (2004), 026008 | DOI | MR

[14] L. Brekke, P. G. O. Freund, “Adic numbers in physics”, Phys. Rep., 233:1 (1993), 1–66 | DOI | MR

[15] N. Moeller, M. Schnabl, “Tachyon condensation in open-closed $p$-adic string theory”, J. High Energy Phys., 2004, no. 1, 011 | DOI | MR | Zbl

[16] V. S. Vladimirov, “O nelineinykh uravneniyakh $p$-adicheskikh otkrytykh, zamknutykh i otkryto-zamknutykh strun”, TMF, 149:3 (2006), 354–367 | MR | Zbl

[17] V. S. Vladimirov, Ya. I. Volovich, “O nelineinom uravnenii dinamiki v teorii $p$-adicheskoi struny”, TMF, 138:3 (2004), 355–368 | MR | Zbl

[18] V. S. Vladimirov, “Ob uravnenii $p$-adicheskoi otkrytoi struny dlya skalyarnogo polya takhionov”, Izv. RAN. Ser. matem, 69:3 (2005), 55–80 | MR | Zbl

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

[20] L. Joukovskaya, Y. Volovich, Energy flow from open to closed strings in a toy model of rolling tachyon, arXiv: math-ph/0308034

[21] Kh. A. Khachatryan, “On solvability of one class of nonlinear integral equations on whole line with a weak singularity at zero”, $p$-Adic Numb., Ultrametric Anal. Appl., 9:4 (2017), 292–305 | DOI | MR | Zbl

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

[23] Kh. A. Khachatryan, “O razreshimosti odnoi granichnoi zadachi v $p$-adicheskoi teorii strun”, Tr. MMO, 79, no. 1, 2018, 117–132 | Zbl

[24] A. Kh. Khachatryan, Kh. A. Khachatryan, “O razreshimosti odnogo nelineinogo integralnogo uravneniya dinamicheskoi teorii struny”, TMF, 195:1 (2018), 44–53 | MR | Zbl

[25] Kh. A. Khachatryan, “O razreshimosti nekotorykh klassov nelineinykh singulyarnykh kraevykh zadach, voznikayuschikh v teorii $p$-adicheskikh otkryto-zamknutykh strun”, TMF, 200:1 (2019), 106–117 | MR | Zbl

[26] Kh. A. Khachatryan, Ts. E. Terdzhyan, M. O. Avetisyan, “Odnoparametricheskoe semeistvo ogranichennykh reshenii dlya odnoi sistemy nelineinykh integralnykh uravnenii na vsei pryamoi”, Izv. NAN Armenii. Matematika, 53:4 (2018), 72–86 | MR | Zbl

[27] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1981 | MR

[28] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, Izd. 3-e, Nauka, M., 1974 | MR

[29] R. Rudin, Funktsionalnyi analiz, Mir, M., 1975

[30] G. M. Fikhtengolts, Kurs differentsialnogo i integralnogo ischisleniya, v. II, Fizmatlit, M., 1970

[31] L. G. Arabadzhyan, N. B. Engibaryan, “Uravneniya v svertkakh i nelineinye funktsionalnye uravneniya”, Itogi nauki i tekhn. Ser. Mat. anal., 22, VINITI, M., 1984, 175–244 | MR

[32] L. G. Arabadzhyan, A. S. Khachatryan, “Ob odnom klasse integralnykh uravnenii tipa svertki”, Matem. sb., 198:7 (2007), 45–62 | MR | Zbl

[33] G. G. Gevorkyan, N. B. Engibaryan, “Novye teoremy dlya integralnogo uravneniya vosstanovleniya”, Izv. NAN Armenii. Matem., 32:1 (1997), 5–20 | MR | Zbl

[34] M. A. Krasnoselskii, P. P. Zabreiko i dr., Integralnye operatory v prostranstvakh summiruemykh funktsii, Nauka, M., 1966 | MR

[35] M. F. Broyan, Kh. A. Khachatryan, “O nekotorykh nelineinykh integralnykh i integrodifferentsialnykh uravneniyakh s nekompaktnymi operatorami na polozhitelnoi polupryamoi”, Ufimsk. matem. zhurnal, 5:2 (2013), 31–42 | MR