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JOURNALS // Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica // Archive

Bul. Acad. Ştiinţe Repub. Mold. Mat., 2022 Number 2, Pages 19–38 (Mi basm570)

On the solubility of a class of two-dimensional integral equations on a quarter plane with monotone nonlinearity

Kh. A. Khachatryana, H. S. Petrosyanb, S. M. Andriyanb

a Yerevan State University, 1, Alex Manoogian St., Yerevan, 0025, Armenia
b Armenian National Agrarian University, 74, Teryan St., Yerevan 0009, Armenia

Abstract: In the paper we study a class of two-dimensional integral equations on a quarter-plane with monotone nonlinearity and substochastic kernel. With specific representations of the kernel and nonlinearity, an equation of this kind arises in various fields of natural science. In particular, such equations occur in the dynamical theory of $p$-adic open-closed strings for the scalar field of tachyons, in the mathematical theory of the geographical spread of a pandemic, in the kinetic theory of gases, and in the theory of radiative transfer in inhomogeneous media.
We prove constructive theorems on the existence of a nontrivial nonnegative and bounded solution. For one important particular case, the existence of a one-parameter family of nonnegative and bounded solutions is also established. Moreover, the asymptotic behavior at infinity of each solution from the given family os studied. At the end of the paper, specific particular examples (of an applied nature) of the kernel and nonlinearity that satisfy all the conditions of the proven statements are given.

Keywords and phrases: two-dimensional equation, nonlinearity, Carathéodory condition, monotonicity, convergence, bounded solution.

MSC: 45G10

Received: 10.04.2022

Language: English

DOI: 10.56415/basm.y2022.i2.p19



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