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On the Solvability of One Class of Two-Dimensional Urysohn Integral Equations

Abstract

We study one class of nonlinear Urysohn integral equations in a quadrant of the plane. It is assumed that, for the corresponding two-dimensional Urysohn operator, some Hammerstein operator with power nonlinearity serves as a minorant in the sense of M. A. Krasnosel’skiĭ.We prove the existence of a nonnegative (nontrivial) and bounded solution for such equations.

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Correspondence to Kh. A. Khachatryan.

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Original Russian Text © Kh.A. Khachatryan, 2017, published in Matematicheskie Trudy, 2017, Vol. 20, No. 2, pp. 193–205.

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Khachatryan, K.A. On the Solvability of One Class of Two-Dimensional Urysohn Integral Equations. Sib. Adv. Math. 28, 166–174 (2018). https://doi.org/10.3103/S1055134418030021

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Keywords

  • Urysohn equation
  • iteration
  • monotonicity
  • power nonlinearity
  • Carathéodory condition