Solvability of a Certain System of Singular Integral Equations with Convex Nonlinearity on the Positive Half-Line


We study a system of nonlinear singular integral equations with a sum-difference kernel on the positive half-line. Such a system occurs (in various representations) in many branches of mathematical physics and applied mathematics. In particular, one encounters a system of equations with a kernel representing a Gaussian distribution and with a power nonlinearity in the dynamic theory of p-adic open-closed strings; in the case when the nonlinearity has a certain exponential structure, such a system occurs in mathematical biology, namely, in the theory of the spatio-temporal distribution of an epidemic.

We prove constructive theorems on the existence of nonnegative nontrivial continuous and bounded solutions. We study the uniqueness and the asymptotic behavior of constructed solutions at infinity. In conclusion, we give concrete applied examples of the mentioned equations that satisfy all assumptions of the proved theorems.

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The authors are grateful to the reviewer for useful remarks.


This work was supported by the Russian Scientific Foundation (project no. 19-11-00223).

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Correspondence to Kh. A. Khachatryan or H. S. Petrosyan.

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Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 1, pp. 31–51.

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Khachatryan, K.A., Petrosyan, H.S. Solvability of a Certain System of Singular Integral Equations with Convex Nonlinearity on the Positive Half-Line. Russ Math. 65, 27–46 (2021).

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  • kernel
  • nonlinearity
  • monotonicity
  • convexity
  • spectral radius
  • limit solutions