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On qualitative properties of the solution of a boundary value problem for a system of nonlinear integral equations

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Abstract

For a system of nonlinear integral equations on the semiaxis, we study a boundary value problem whose matrix kernel has unit spectral radius. This boundary value problem has applications in various areas of physics and biology. In particular, such problems arise in the dynamical theory of \(p\)-adic strings for the scalar field of tachyons, in the mathematical theory of spread of epidemic diseases, in the kinetic theory of gases, and in the theory of radiative transfer. The questions of the existence, absence, and uniqueness of a nontrivial solution of this boundary value problem are discussed. In particular, it is proved that a boundary value problem with a zero boundary conditions at infinity has only a trivial solution in the class of nonnegative and bounded functions. It is also proved that if at least one of the values at infinity is positive, then this problem has a convex nontrivial nonnegative bounded and continuous solution. At the end of this paper, examples of the matrix kernel and nonlinearity are provided that satisfy all the conditions of the proved theorems.

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Acknowledgments

The authors thank the referee for the useful remarks.

Funding

The research by the first author was conducted under the support of the Science Committee, Republic of Armenia, within research project 23RL-1A027. The research by the second author was conducted under the support of the Science Committee, Republic of Armenia, within research project 21T-1A047.

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Authors and Affiliations

  1. Faculty of Mathematics and Mechanics, Yerevan State University, Yerevan, Armenia

    Kh. A. Khachatryan

  2. Department of Higher Mathematics, Armenian National Agrarian University, Yerevan, Armenia

    H. S. Petrosyan

Authors
  1. Kh. A. Khachatryan
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  2. H. S. Petrosyan
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Correspondence to Kh. A. Khachatryan.

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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2024, Vol. 218, pp. 168–186 https://doi.org/10.4213/tmf10525.

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Khachatryan, K.A., Petrosyan, H.S. On qualitative properties of the solution of a boundary value problem for a system of nonlinear integral equations. Theor Math Phys 218, 145–162 (2024). https://doi.org/10.1134/S0040577924010100

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  • DOI: https://doi.org/10.1134/S0040577924010100

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