On the solubility of certain classes of non-linear integral equations

Kh. A. Khachatryan

Izvestiya. Mathematics , Tome 82 (2018) no. 2, pp. 407-427.

Résumé

We study classes of non-linear integral equations that have immediate application to $p$-adic mathematical physics and to cosmology. We prove existence and uniqueness theorems for non-trivial solutions in the space of bounded functions.

Détail

Keywords: $p$-adic string, non-linear equation, iterations, monotonicity of a solution, limit of a solution.

Bibliographie

[1] Ya. I. Volovich, “Some properties of dynamical equations in $p$-adic string and SFT models”, Proc. Steklov Inst. Math., 245 (2004), 281–288 | MR | Zbl

[2] N. Barnaby, “Caustic formation in tachyon effective field theories”, J. High Energy Phys., 2004, no. 7, 025, 23 pp. | DOI | MR

[3] V. S. Vladimirov, Ya. I. Volovich, “Nonlinear dynamics equation in $p$-adic string theory”, Theoret. and Math. Phys., 138:3 (2004), 297–309 | DOI | DOI | MR | Zbl

[4] P. Górka, E. G. Reyes, “Sobolev spaces on locally compact abelian groups and the bosonic string equation”, J. Aust. Math. Soc., 98:1 (2015), 39–53 | DOI | MR | Zbl

[5] I. Ya. Aref'eva, “Rolling tachyon on non-BPS branes and $p$-adic strings”, Proc. Steklov Inst. Math., 245 (2004), 40–47 | MR | Zbl

[6] I. Ya. Aref'eva, I. V. Volovich, “The null energy condition and cosmology”, Theoret. and Math. Phys., 155:1 (2008), 503–511 | DOI | DOI | MR | Zbl

[7] I. Ya. Aref'eva, I. V. Volovich, “Cosmological daemon”, J. High Energy Phys., 2011:8 (2011), 102, 29 pp. | DOI | Zbl

[8] L. V. Joukovskaya, “Iterative method for solving nonlinear integral equations describing rolling solutions in string theory”, Theoret. and Math. Phys., 146:3 (2006), 335–342 | DOI | DOI | MR | Zbl

[9] 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), 34–41 | DOI

[10] V. S. Vladimirov, “Solutions of $p$-adic string equations”, Theoret. and Math. Phys., 167:2 (2011), 539–546 | DOI | DOI | MR | Zbl

[11] M. A. Krasnosel'skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964, 381 pp. | MR | MR | Zbl | Zbl

[12] M. A. Krasnosel'skii, P. P. Zabreiko, E. I. Pustyl'nik, P. E. Sobolevskii, Integral operators in spaces of summable functions, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff International Publishing, Leiden, 1976, xv+520 pp. | MR | MR | Zbl | Zbl

[13] A. Kh. Khachatryan, Kh. A. Khachatryan, “On an integral equation with monotonic nonlinearity”, Mem. Differential Equations Math. Phys., 51 (2010), 59–72 | MR | Zbl

[14] L. G. Arabadzhyan, N. B. Engibaryan, “Convolution equations and nonlinear functional equations”, J. Soviet Math., 36:6 (1987), 745–791 | DOI | MR | Zbl

[15] I. P. Natanson, Theory of functions of a real variable, v. 1, 2, Frederick Ungar Publishing Co., New York, 1955, 1961, 277 pp., 265 pp. | MR | MR | MR | Zbl

[16] G. G. Gevorkyan, N. B. Engibaryan, “New theorems for the renewal integral equation”, J. Contemp. Math. Anal., 32:1 (1997), 2–16 | MR | Zbl

[17] A. N. Kolmogorov, S. V. Fomin, Elements of the theory of functions and functional analysis, v. I, II, Graylock Press, Albany, N.Y., 1957, 1961, ix+129, ix+128 pp. | MR | MR | MR | Zbl | Zbl