On nonlinear convolution-type integral equations in the theory

A. Kh. Khachatryan ; Kh. A. Khachatryan ; H. S. Petrosyan

Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 1, pp. 184-200

Résumé

We study a class of integral equations of convolution type on the whole line with a monotone and odd nonlinearity. We prove constructive existence and absence theorems for nonnegative (nontrivial) and bounded solutions. We study the asymptotic behavior of the constructed solution at $\pm\infty$. We also prove the uniqueness of the solution in the class of nonnegative (nonzero) and bounded functions and present specific examples of this class of equations that can be applied in various fields of mathematical physics.

Détail

Keywords: monotonicity, nonlinearity, convexity
Mots-clés : kernel, nonnegative solution, convolution.

Bibliographie

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

[2] I. Ya. Aref'eva, B. G. Dragovic, 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

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

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

[5] V. S. Vladimirov, “O resheniyakh $p$-adicheskikh strunnykh uravnenii”, TMF, 167:2 (2011), 163–170 | DOI | DOI | MR

[6] V. S. Vladimirov, “Matematicheskie voprosy teorii nelineinykh psevdodifferentsialnykh uravnenii $p$-adicheskikh strun”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1:22 (2011), 34–31 | DOI

[7] V. S. Vladimirov, “The equation of the $p$-adic closed strings for the scalar tachyon field”, Sci. China Ser. A, 51:4 (2008), 754–764 | DOI | MR

[8] V. S. Vladimirov, “K voprosu ob asimptotike pri $|t|\rightarrow\infty $ reshenii kraevykh zadach dlya $p$-adicheskikh strun”, TMF, 157:3 (2008), 325–333 | DOI | DOI | MR | Zbl

[9] V. S. Vladimirov, “On the equations for $p$-adic closed and open strings”, $p$-Adic Numbers Ultrametric Anal. Appl., 1:1 (2009), 79–87 | DOI | MR

[10] V. S. Vladimirov, “O nelineinykh uravneniyakh $p$-adicheskikh strun dlya skalyarnykh polei takhionov. Izbrannye voprosy matematicheskoi fiziki i $p$-adicheskogo analiza”, Trudy MIAN, 265, MAIK “Nauka/Interperiodika”, M., 2009, 254–272 | DOI | MR | Zbl

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

[12] V. S. Vladimirov, “K voprosu necuschestvovaniya reshenii uravnenii $p$-adicheskikh strun”, TMF, 174:2 (2013), 208–215 | DOI | DOI | MR | Zbl

[13] 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”, JHEP, 09 (2003), 012, 15 pp., arXiv: hep-th/0301137 | DOI | MR

[14] L. Joukovskaya, “Dynamics in nonlocal cosmological models derived from string field theory”, Phys. Rev. D, 76:10 (2007), 105007, 12 pp., arXiv: 0707.1545 | DOI | MR

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

[16] O. Diekmann, H. G. Kaper, “On the bounded solutions of a nonlinear convolution equation”, Nonlinear Anal., 2:6 (1978), 721–737 | DOI | MR

[17] A. G. Sergeev, Kh. A. Khachatryan, “O razreshimosti odnogo klassa nelineinykh integralnykh uravnenii v zadache rasprostraneniya epidemii”, Tr. MMO, 80, no. 1, MTsNMO, M., 2019, 113–131 | DOI | MR

[18] C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67, Springer, New York, 1988 | DOI | MR

[19] V. V. Sobolev, “Problema Milna dlya neodnorodnoi atmosfery”, Dokl. AN SSSR, 239:3 (1978), 558–561 | MR

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

[21] R. Z. Zhdanov, “Separation of variables in the nonlinear wave equation”, J. Phys. A, 27:9 (1994), 1291–1297 | DOI | MR

[22] A. M. Grundland, E. Infeld, “A family of non-linear Klein–Gordon equations and theis solutions”, J. Math. Phys., 33:7 (1992), 2498–2503 | DOI | MR

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

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

[25] Kh. A. Khachatryan, “Suschestvovanie i edinstvennost resheniya odnoi granichnoi zadachi dlya integralnogo uravneniya svertki s monotonnoi nelineinostyu”, Izv. RAN. Ser. matem., 84:4 (2020), 198–207 | DOI | DOI | MR

[26] U. Rudin, Funktsionalnyi analiz, Mir, M., 1975 | MR

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

[28] G. G. Khardi, Dzh. E. Littlvud, G. Polia, Neravenstva, IL, M., 1948 | MR | Zbl

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

[30] I. M. Gelfand, G. E. Shilov, Obobschennye funktsii, v. 2, Prostranstva osnovnykh i obobschennykh funktsii, Fizmatgiz, M., 1958 | MR | MR

[31] A. Kh. Khachatryan, Kh. A. Khachatryan, “Solvability of a class of nonlinear pseudo-differential equations in $\mathbb{R}^n$”, $p$-Adic Numbers Ultrametric Anal. Appl., 10:2 (2018), 90–99 | DOI | MR

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