Abstract
We consider a nonlinear integro-differential equation with zero operator in the differential part whose integral operator contains several rapidly varying kernels. This paper continues the research carried out earlier for equations with only one rapidly varying kernel. We prove that the conditions for the solvability of the corresponding iteration problems, as in the linear case, are not differential (as in problems with nonzero operator in the differential part) but rather integro-differential equations, and the structure of these equations is substantially influenced by the nonlinearity. In the nonlinear case, so-called resonances can arise, significantly complicating the development of the corresponding algorithm of the regularization method. The paper deals with the nonresonant case.
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Notes
The function \(f(y,t) \) is taken in the form of a polynomial to simplify the calculations. It can be assumed that \(f(y,t)\) is analytic in \(y \); i.e., \(N=\infty \) in item 4.
Here and throughout the following, by \((\thinspace ,) \) we denote the standard inner product on the complex space \( {\mathbb C}^{3}\).
The bullet stands for differentiation with respect to \(t\).
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Translated by V. Potapchouck
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Bobodzhanov, A.A., Bobodzhanova, M.A. & Safonov, V.F. Regularized Asymptotic Solutions of Nonlinear Integro-Differential Equations with Zero Operator in the Differential Part and with Several Rapidly Varying Kernels. Diff Equat 57, 753–767 (2021). https://doi.org/10.1134/S0012266121060057
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DOI: https://doi.org/10.1134/S0012266121060057