Geodesic
Constructive study of the solvability of one class of nonlinear integral equations with a symmetric kernel
Matematičeskie trudy, Tome 27 (2024) no. 3, pp. 111-138 Cet article a éte moissonné depuis la source Math-Net.Ru

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A class of nonlinear integral equations of Hammerstein type with a symmetric and substochastic kernel is considered. This class of equations is encountered in many branches of physics and mathematical biology. In particular, equations of this nature arise in the dynamic theory of a $p$-adic string, in the kinetic theory of gases, and in various model problems of the mathematical theory of the spread of epidemic diseases. A constructive theorem on the existence of a nontrivial bounded nonnegative continuous and monotonically nondecreasing solution is proved. In the class of nontrivial nonnegative and bounded functions, the uniqueness of the solution is also proved. The results obtained are used to study a nonlinear integral equation on the entire line with an almost difference kernel. At the end of the paper, particular examples of the kernel and nonlinearity are given that are of applied nature in the above theories. 
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Kh. A. Khachatryan; H. S. Petrosyan. Constructive study of the solvability of one class of nonlinear integral equations with a symmetric kernel. Matematičeskie trudy, Tome 27 (2024) no. 3, pp. 111-138. http://geodesic.mathdoc.fr/item/MT_2024_27_3_a6/

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

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

[3] Vladimirov V. S., “The equation of the p-adic open string for the scalar tachyon field”, Izv. Math., 69:3 (2005), 487–512 | DOI | DOI | MR

[4] Zhukovskaya L. V., “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

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

[6] Khachatryan A. Kh., Khachatryan Kh. A., “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 | DOI | MR

[7] Khachatryan A. Kh., Khachatryan Kh. A., “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 | DOI | MR | MR

[8] Atkinson C., Reuter G. E. H., “Deterministic epidemic waves”, Math. Proc. Camb. Phil. Soc., 80 (1976), 315–330 | DOI | MR

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

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

[11] Aref'eva I. Ya., Volovich I. V., “On nonlocal cosmological equations on half-line”, Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 2011, no. 1, 16–27 (Russian) | DOI

[12] Aref'eva I. Ya., Volovich I. V., “Cosmological daemon”, JHEP, 2011:8 (2011), 1–32 | DOI

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

[14] Khachatryan Kh. A., “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), 198–207 | DOI | MR

[15] Khachatryan Kh. A., Petrosyan H. S., “On qualitative properties of the solution of a nonlinear boundary value problem in the dynamic theory of p-adic strings”, Vestn. St. Petersburg Univ., Ser. 10, Prikl. Mat. Inform. Prots. Upr., 16 (2020), 423–436 (in Russian) | MR

[16] Petrosyan H. S., Khachatryan Kh. A., “Uniqueness of the Solution of a Class of Integral Equations with Sum-Difference. Kernel and with Convex Nonlinearity on the Positive Half-Line”, Math. Notes, 113:4 (2023), 529–543 | DOI

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

[18] Kolmogorov A. N., Fomin S. V., Introductory real analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1970 | MR

[19] Rudin W., Real and Complex Analysis, Third Edition, McGraw-Hill Inc., 1987 | MR