A system of integral equations on the entire axis with convex and monotone nonlinearity. (English. Russian original) Zbl 1503.45003
J. Contemp. Math. Anal., Armen. Acad. Sci. 57, No. 5, 311-322 (2022); translation from Izv. Nats. Akad. Nauk Armen., Mat. 57, No. 5, 65-80 (2022).
Summary: A system of nonlinear convolution-type integral equations on the entire axis is considered in this work. This system is met in many areas of mathematical physics and mathematical biology. The issues of existence and absence of nontrivial bounded solutions are investigated. The theorem about the uniqueness of the solution in a certain class of bounded functions is proved. In the end of the work, applied examples of this system are provided.
MSC:
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 45G15 | Systems of nonlinear integral equations |
References:
| [1] | Lancaster, P., Theory of Matrices (1969), New York: Academic Press, New York · Zbl 0186.05301 |
| [2] | Brekke, L.; Freund, P. G. O.; Olson, M.; Witten, E., Non-archimedean string dynamics, Nucl. Phys. B, 302, 365-402 (1988) · doi:10.1016/0550-3213(88)90207-6 |
| [3] | Vladimirov, V. S.; Volovich, Ya. I., Nonlinear dynamics equation in \(p\), Theor. Math. Phys., 138, 297-309 (2004) · Zbl 1178.81174 · doi:10.1023/B:TAMP.0000018447.02723.29 |
| [4] | Engibaryan, N. B., On a problem in nonlinear radiative transfer, Astrofizika, 2, 31-36 (1966) |
| [5] | C. Cercignani, The Boltzmann Equation and Its Application, Applied Mathematical Sciences, Vol. 67 (Springer, New York, 1988). https://doi.org/10.1007/978-1-4612-1039-9 · Zbl 0646.76001 |
| [6] | Khachatryan, A. Kh.; Khachatryan, Kh. A., Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave, Theor. Math. Phys., 189, 1609-1623 (2016) · Zbl 1359.76177 · doi:10.1134/S0040577916110064 |
| [7] | Diekmann, O., Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 6, 109-130 (1978) · Zbl 0415.92020 · doi:10.1007/BF02450783 |
| [8] | Diekmann, O., Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differ. Equation, 33, 58-73 (1979) · Zbl 0377.45007 · doi:10.1016/0022-0396(79)90080-9 |
| [9] | Khachatryan, Kh. A., “The solvability of a system of nonlinear integral equations of Hammerstein type on the whole line,” Izv. Saratov. Univ. Nov. Ser. Mat., Mekh, Inf., 19, 164-181 (2019) · Zbl 1434.45004 |
| [10] | Diekmann, O.; Kaper, H. G., “On the bounded solutions of a nonlinear convolution equation,” Nonlinear Anal.: Theory, Methods Appl., 2, 721-737 (1978) · Zbl 0433.92028 |
| [11] | Vladimirov, V. S., Solutions of \(p\), Theor. Math. Phys., 167, 539-546 (2011) · Zbl 1274.81198 · doi:10.1007/s11232-011-0040-z |
| [12] | Joukovskaya, L. V., Iterative method for solving nonlinear integral equations describing rolling solutions in string theory, Theor. Math. Phys., 146, 335-342 (2006) · Zbl 1177.81113 · doi:10.1007/s11232-006-0043-3 |
| [13] | Khachatryan, Kh. A., On the solubility of certain classes of non-linear integral equations in \(p\), Izv. Math., 82, 407-427 (2018) · Zbl 1395.45010 · doi:10.1070/IM8580 |
| [14] | Khachatryan, Kh. A., On the solvability of a boundary value problem in \(p\), Trans. Moscow Math. Soc., 2018, 101-115 (2018) · Zbl 1411.45004 · doi:10.1090/mosc/281 |
| [15] | Khachatryan, Kh. A.; Petrosyan, H. S., Integral equations on the whole line with monotone nonlinearity and difference kernel, J. Math. Sci., 255, 790-804 (2021) · Zbl 1465.45006 · doi:10.1007/s10958-021-05416-0 |
| [16] | Khachatryan, Kh. A., On some systems of nonlinear integral Hammerstein-type equations on the semiaxis, Ukr. Math. J., 62, 630-647 (2010) · Zbl 1224.45014 · doi:10.1007/s11253-010-0376-9 |
| [17] | Khachatryan, Kh. A.; Terdzhyan, Ts. E.; Sardanyan, T. G., On the solvability of one system of nonlinear Hammerstein-type integral equations on the semiaxis, Ukr. Math. J., 69, 1287-1305 (2018) · Zbl 1423.45003 · doi:10.1007/s11253-017-1431-6 |
| [18] | Kolmogorov, A. N.; Fomin, S. V., Elements of the Theory of Functions and Functional Analysis (1957) |
| [19] | Rudin, W., Functional Analysis (1973), New York: McGraw-Hill, New York · Zbl 0253.46001 |
| [20] | L. G. Arabadzhyan and N. B. Engibaryan, ‘‘Convolution equations and nonlinear functional equations,’’ in Results of Science and Engineering: Mathematical Analysis (Vsesoyuznyi Inst. Nauchn. Tekh. Informatsii, Moscow, 1984), Vol. 22, pp. 175-244. · Zbl 0568.45004 |
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