Abstract
The method of weighted metrics in a cone of the space of continuous functions is used to prove a global theorem on the existence and uniqueness of a nonnegative nontrivial solution for a system of integro-differential equations of convolution type with power nonlinearity. We demonstrate that the solution can be found by the method of successive approximations of the Picard type. We also obtain some exact a priori estimates for solution.
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REFERENCES
S. N. Askhabov, N. K. Karapetyants, and A. Ya. Yakubov, “Integral Equations of Convolution Type with Power Nonlinearity and Systems of Such Equations,” Dokl. Akad. Nauk SSSR 311 (5), 1035–1039 (1990) [Dokl. Math. 41 (2), 323–327 (1990)].
S. N. Askhabov and M. A. Betilgiriev, “Nonlinear Integral Equations of Convolution Type with Almost Increasing Kernels in Cones,” Differentsial’nye Uravneniya 27 (2), 321–330 (1991) [Differential Equations 27 (2), 234–242 (1991)].
S. N. Askhabov, Nonlinear Convolution Type Equations (Fizmatlit, Moscow, 2009) [in Russian].
W. Okrasinski, “On the Existence and Uniqueness of Nonnegative Solutions of a Certain Nonlinear Convolution Equation,” Ann. Polon. Math. 36 (1), 61–72 (1976).
W. Okrasinski, “Nonlinear Volterra Equations and Physical Applications,” Extracta Math. 4 (2), 51–74 (1989).
J. J. Keller, “Propagation of Simple Nonlinear Waves in Gas Filled Tubes with Friction,” Z. Angew. Math. Phys. 32 (2), 170–181 (1981).
A. N. Tikhonov, “On the Cooling of Bodies by Radiation Following the Stefan–Boltzmann Law,” Izv. Akad. Nauk SSSR, Otdel. Mat. i Estestv. Nauk, Ser. Geogr. Geophys. No. 3, 461–479 (1937).
G. B. Ermentrout and J. D. Cowan, “Secondary Bifurcation in Neuronal Nets,” SIAM J. Appl. Math. 39 (2), 323–340 (1980).
L. V. Wolfersdorf, “Einige Klassen quadratischer Integralgleichungen,” Sitz. Sachs. Akad. Wiss. Leipzig. Math.-Naturwiss. Klasse 128 (2), 1–34 (2000).
H. Brunner, Volterra Integral Equations: An Introduction to the Theory and Applications (Univ. Press, Cambridge, 2017).
R. E. Edwards, Functional Analysis: Theory and Applications (Holt, Rinehart, and Winston, New York, 1965; Mir, Moscow, 1969).
S. N. Askhabov, “Integro-Differential Equation of Convolution Type with a Power Nonlinearity and an Inhomogeneity in the Linear Part,” Differentsial’nye Uravneniya 56 (6), 786–795 (2020) [Differential Equations 56 (6), 775–784 (2020)].
S. N. Askhabov and N. K. Karapetyants, “Discrete Equations of Convolution Type with Monotone Nonlinearity,” Differentsial’nye Uravneniya 25 (10), 1777–1784 (1989) [Differential Equations 25 (10), 1255–1261 (1989)].
S. N. Askhabov and N. K. Karapetyants, “Convolution Type Discrete Equations with Monotonous Nonlinearity in Complex Spaces,” J. Integral Equat. Math. Phys. 1 (1), 44–66 (1992).
S. N. Askhabov and N. K. Karapetyants, “Discrete Equations of Convolution Type with Monotone Nonlinearity in Complex Spaces,” Dokl. Akad. Nauk 322 (6), 1015–1018 (1992) [Dokl. Math. 45 (1), 206–210 (1992)].
Funding
The author was supported by the Russian Foundation for Basic Research (project no. 18–41–200001) and the State Task on the Project “Nonlinear Singular Integro-Differential Equations and Boundary Value Problems” (Agreement no. 075–03–2021–071, December 29, 2020).
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Translated by L.B. Vertgeim
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Askhabov, S.N. A System of Integro-Differential Equations of Convolution Type with Power Nonlinearity. J. Appl. Ind. Math. 15, 365–375 (2021). https://doi.org/10.1134/S1990478921030017
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DOI: https://doi.org/10.1134/S1990478921030017