On non-trivial solvability of one system of non-linear integral equations on the real axis

Kh. A. Khachatryan ; H. S. Petrosyan

Izvestiya. Mathematics , Tome 87 (2023) no. 5, pp. 1062-1077.

Résumé

A system of singular integral equations with monotonic and convex non-linearity on the entire real line is considered. System of this form have applications in many areas of natural science. In particular, such systems arise in the theory of $p$-adic open-closed strings, in the mathematical theory of spatial-temporal epidemic spread within the framework of the well known Diekmann–Kaper model, in the kinetic theory of gases, in the radiative transfer theory. An existence theorem for a non-trivial and bounded solution is proved. The asymptotic behaviour of the constructed solution at $\pm\infty$ is also studied. Specific examples of non-linearities and kernel functions having an applied character are given.

Détail

Keywords: convexity, monotonicity, spectral radius, non-linearity, bounded solution.

Bibliographie

[1] P. Lancaster, Theory of matrices, Academic Press, New York–London, 1969 | MR | Zbl

[2] V. S. Vladimirov and Ya. I. Volovich, “Nonlinear Dynamics Equation in $p$-Adic String Theory”, Theoret. and Math. Phys., 138:3 (2004), 297–309 | DOI

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

[4] Kh. A. Khachatryan, “On the solvability of a boundary value problem in $ p$-adic string theory”, Trans. Moscow Math. Soc., 2018, 101–115 | DOI | MR

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

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

[7] A. Kh. Khachatryan and Kh. A. Khachatryan, “Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave”, Theoret. and Math. Phys., 189:2 (2016), 1609–1623 | DOI

[8] C. Cercignani, The Boltzmann equation and its applications, Appl. Math. Sci., 67, Springer-Verlag, New-York, 1988 | DOI | MR | Zbl

[9] N. B. Engibaryan, “On a problem in nonlinear radiative transfer”, Astrophysics, 2:1 (1966), 12–14 | DOI

[10] Kh. A. Khachatryan, “The solvability of a system of nonlinear integral equations of Hammerstein type on the whole line”, Izv. Saratov Univ. Math. Mech. Inform., 19:2 (2019), 164–181 (Russian) | DOI | MR | Zbl

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

[12] Kh. A. Khachatryan, “On the solubility of certain classes of non-linear integral equations in $p$-adic string theory”, Izv. Math., 82:2 (2018), 407–427 | DOI

[13] Kh. A. Khachatryan, “Existence and uniqueness of solution of a certain boundary-value problem for a convolution integral equation with monotone non-linearity”, Izv. Math., 84:4 (2020), 807–815 | DOI

[14] Kh. A. Khachatryan and 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 | Zbl

[15] A. N. Kolmogorov and S. V. Fomin, Introductory real analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1970 | MR | Zbl

[16] W. Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York–Düsseldorf–Johannesburg, 1973 | MR | Zbl

[17] N. B. Engibaryan, “Conservative systems of integral convolution equations on the half-line and the entire line”, Sb. Math., 193:6 (2002), 847–867 | DOI

[18] L. G. Arabadzhyan and A. S. Khachatryan, “A class of integral equations of convolution type”, Sb. Math., 198:7 (2007), 949–966 | DOI